{"id":21879,"date":"2024-01-04T11:59:35","date_gmt":"2024-01-04T06:29:35","guid":{"rendered":"https:\/\/apboardsolutions.in\/?p=21879"},"modified":"2024-01-05T10:59:27","modified_gmt":"2024-01-05T05:29:27","slug":"inter-1st-year-maths-1a-products-of-vectors-solutions-ex-5c","status":"publish","type":"post","link":"https:\/\/apboardsolutions.in\/inter-1st-year-maths-1a-products-of-vectors-solutions-ex-5c\/","title":{"rendered":"Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c)"},"content":{"rendered":"

Practicing the Intermediate 1st Year Maths 1A Textbook Solutions<\/a> Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(c) will help students to clear their doubts quickly.<\/p>\n

Intermediate 1st Year Maths 1A Products of Vectors Solutions Exercise 5(c)<\/h2>\n

I.<\/span><\/p>\n

Question 1.
\nCompute \\([\\overline{\\mathbf{i}}-\\overline{\\mathbf{j}} \\overline{\\mathbf{j}}-\\overline{\\mathbf{k}} \\overline{\\mathbf{k}}-\\overline{\\mathbf{i}}]\\)
\nSolution:
\n\"Inter<\/p>\n

Question 2.
\nIf \\(\\overline{\\mathbf{a}}=\\overline{\\mathbf{i}}-2 \\overline{\\mathbf{j}}-3 \\overline{\\mathbf{k}}, \\overline{\\mathbf{b}}=2 \\overline{\\mathbf{i}}+\\overline{\\mathbf{j}}-\\overline{\\mathbf{k}}\\), \\(\\bar{c}=\\bar{i}+3 \\bar{j}-2 \\bar{k}\\), then compute \\(\\overline{\\mathbf{a}} \\cdot(\\overline{\\mathbf{b}} \\times \\overline{\\mathbf{c}})\\).
\nSolution:
\n\"Inter<\/p>\n

\"Inter<\/p>\n

Question 3.
\nIf \\(\\bar{a}\\) = (1, -1, -6), \\(\\bar{b}\\) = (1, -3, 4) and \\(\\bar{c}\\) = (2, -5, 3), then compute the following
\n(i) \\(\\overline{\\mathbf{a}} \\cdot(\\bar{b} \\times \\bar{c})\\)
\n(ii) \\(\\overline{\\mathbf{a}} \\times(\\overline{\\mathbf{b}} \\times \\overline{\\mathbf{c}})\\)
\n(iii) \\((\\overline{\\mathbf{a}} \\times \\overline{\\mathbf{b}}) \\times \\overline{\\mathbf{c}}\\)
\nSolution:
\n\"Inter
\n\"Inter<\/p>\n

Question 4.
\nSimplify the following.
\n(i) \\((\\bar{i}-2 \\bar{j}+3 \\bar{k}) \\times(2 i+j-\\bar{k}) \\cdot(\\bar{j}+\\bar{k})\\)
\n(ii) \\((2 \\bar{i}-3 \\bar{j}+\\bar{k}) \\cdot(\\bar{i}-\\bar{j}+2 \\bar{k}) \\cdot(2 \\bar{i}+\\bar{j}+\\bar{k})\\)
\nSolution:
\n\"Inter<\/p>\n

Question 5.
\nFind the volume of the parallelopiped having coterminous edges.
\n\\(\\overline{\\mathbf{i}}+\\overline{\\mathbf{j}}+\\overline{\\mathbf{k}}, \\overline{\\mathbf{i}}-\\overline{\\mathbf{j}}\\) and \\(\\overline{\\mathbf{i}}+\\mathbf{2} \\overline{\\mathbf{j}}-\\overline{\\mathbf{k}}\\)
\nSolution:
\nLet \\(\\overline{\\mathrm{a}}=\\overline{\\mathrm{i}}+\\overline{\\mathrm{j}}+\\overline{\\mathrm{k}}, \\overline{\\mathrm{b}}=\\overline{\\mathrm{i}}-\\overline{\\mathrm{j}}\\) and \\(\\overline{\\mathrm{c}}=\\overline{\\mathrm{i}}+2 \\overline{\\mathrm{j}}-\\overline{\\mathrm{k}}\\)
\nVolume of the parallelopiped = \\([(\\bar{a} \\bar{b} \\bar{c})]\\)
\n= \\(\\left|\\begin{array}{ccc}
\n1 & 1 & 1 \\\\
\n1 & -1 & 0 \\\\
\n1 & 2 & -1
\n\\end{array}\\right|\\)
\n= 1(1 – 0) – 1(-1 – 0) + 1(2 + 1)
\n= 1 + 1 + 3
\n= 5 cubic units.<\/p>\n

Question 6.
\nFind t for which the vectors \\(\\mathbf{2} \\overline{\\mathbf{i}}-\\mathbf{3} \\overline{\\mathbf{j}}+\\overline{\\mathbf{k}}\\), \\(\\bar{i}+2 \\mathbf{j}-3 \\bar{k}\\) and \\(\\bar{j}-t \\bar{k}\\) are coplanar.
\nSolution:
\n\"Inter<\/p>\n

\"Inter<\/p>\n

Question 7.
\nFor non-coplanar vectors, \\(\\bar{a}\\), \\(\\bar{b}\\) and \\(\\bar{c}\\) determine p for which the vector \\(\\overline{\\mathbf{a}}+\\overline{\\mathbf{b}}+\\overline{\\mathbf{c}}, \\overline{\\mathbf{a}}+\\mathbf{p} \\overline{\\mathbf{b}}+\\mathbf{2} \\overline{\\mathbf{c}}\\) and \\(-\\overline{\\mathbf{a}}+\\overline{\\mathbf{b}}+\\overline{\\mathbf{c}}\\) are coplanar.
\nSolution:
\n\"Inter<\/p>\n

Question 8.
\nDetermine \u03bb, for which the volume of the parallelopiped having coterminous edges \\(\\bar{i}+\\bar{j}\\), \\(3 \\overline{\\mathbf{i}}-\\overline{\\mathbf{j}}\\) and \\(3 \\bar{j}+\\lambda \\bar{k}\\) is 16 cubic units.
\nSolution:
\n\"Inter<\/p>\n

Question 9.
\nFind the volume of the tetrahedron having the edges \\(\\overline{\\mathbf{i}}+\\overline{\\mathbf{j}}+\\overline{\\mathbf{k}}, \\quad \\mathbf{i}-\\overline{\\mathbf{j}}\\) and \\(\\bar{i}+2 \\bar{j}+\\bar{k}\\).
\nSolution:
\n\"Inter<\/p>\n

Question 10.
\nLet \\(\\bar{a}\\), \\(\\bar{b}\\) and \\(\\bar{c}\\) be non-coplanar vectors and \\(\\bar{\\alpha}=\\bar{a}+2 \\bar{b}+3 c, \\quad \\bar{\\beta}=2 \\bar{a}+\\bar{b}-2 c\\) and \\(\\bar{\\gamma}=3 \\bar{a}-7 \\bar{c}\\), then find \\(\\left[\\begin{array}{lll}
\n\\bar{\\alpha} & \\bar{\\beta} & \\bar{\\gamma}
\n\\end{array}\\right]\\).
\nSolution:
\n\"Inter<\/p>\n

Question 11.
\nLet \\(\\bar{a}\\), \\(\\bar{b}\\) and \\(\\bar{c}\\) be non-coplanar vectors. If \\(|2 \\bar{a}-\\bar{b}+3 \\bar{c}|, \\bar{a}+\\bar{b}-2 \\bar{c},|\\bar{a}+\\bar{b}-3 \\bar{c}|\\) = \\(\\lambda[\\overline{\\mathbf{a}} \\overline{\\mathbf{b}} \\overline{\\mathbf{c}}]\\), then find the value of \u03bb.
\nSolution:
\n\"Inter
\n\"Inter<\/p>\n

Question 12.
\nLet \\(\\bar{a}\\), \\(\\bar{b}\\) and \\(\\bar{c}\\) be non-coplanar vectors, if \\(\\left[\\begin{array}{lll}
\n\\bar{a}+2 \\bar{b} & 2 \\bar{b}+\\bar{c} & 5 \\bar{c}+\\bar{a}
\n\\end{array}\\right]\\) = \\(\\lambda\\left[\\begin{array}{lll}
\n\\bar{a} & \\bar{b} & \\bar{c}
\n\\end{array}\\right]\\), then find \u03bb.
\nSolution:
\n\"Inter<\/p>\n

\"Inter<\/p>\n

Question 13.
\nIf \\(\\bar{a}\\), \\(\\bar{b}\\) and \\(\\bar{c}\\) are non-coplanar vectors, then find the value of \\(\\frac{(\\bar{a}+2 \\bar{b}-\\bar{c}) \\cdot[(\\bar{a}-\\bar{b}) \\times(\\bar{a}-\\bar{b}-\\bar{c})]}{[\\bar{a} \\bar{b} \\bar{c}]}\\)
\nSolution:
\n\"Inter<\/p>\n

Question 14.
\nIf \\(\\bar{a}\\), \\(\\bar{b}\\) and \\(\\bar{c}\\) are mutually perpendicular unit vectors, then find the value of \\(\\left[\\begin{array}{lll}
\n\\bar{a} & \\bar{b} & \\bar{c}
\n\\end{array}\\right]^{2}\\).
\nSolution:
\n\"Inter<\/p>\n

Question 15.
\n\\(\\overline{\\mathbf{a}}, \\overline{\\mathbf{b}}, \\overline{\\mathbf{c}}\\) are non-zero vectors and \\(\\overline{\\mathbf{a}}\\) is perpendicular to both \\(\\overline{\\mathbf{b}}\\) and \\(\\overline{\\mathbf{c}}\\). If \\(|\\overline{\\mathbf{a}}|\\) = 2, \\(|\\overline{\\mathbf{b}}|\\) = 3, \\(|\\overline{\\mathbf{c}}|\\) = 4 and \\((\\bar{b}, \\bar{c})=\\frac{2 \\pi}{3}\\), then find \\(|[\\bar{a} \\bar{b} \\bar{c}]|\\).
\nSolution:
\n\"Inter<\/p>\n

Question 16.
\nIf \\(\\overline{\\mathbf{a}}, \\overline{\\mathbf{b}}, \\overline{\\mathbf{c}}\\) are unit coplanar vectors, then find \\(\\left[\\begin{array}{lll}
\n2 \\bar{a}-\\bar{b} & 2 \\bar{b}-\\bar{c} & 2 \\bar{c}-\\bar{a}
\n\\end{array}\\right]\\)
\nSolution:
\n\"Inter<\/p>\n

II.<\/span><\/p>\n

Question 1.
\nIf \\(\\left[\\begin{array}{lll}
\n\\bar{b} & \\bar{c} & \\bar{d}
\n\\end{array}\\right]+\\left[\\begin{array}{lll}
\n\\bar{c} & \\bar{a} & \\bar{d}
\n\\end{array}\\right]+\\left[\\begin{array}{lll}
\n\\bar{a} & \\bar{b} & \\bar{d}
\n\\end{array}\\right]\\) = \\(\\left[\\begin{array}{lll}
\n\\overline{\\mathbf{a}} & \\overline{\\mathbf{b}} & \\overline{\\mathbf{c}}
\n\\end{array}\\right]\\) then show that the points with position vectors \\(\\overline{\\mathbf{a}}, \\overline{\\mathbf{b}}, \\overline{\\mathbf{c}}\\) and \\(\\bar{d}\\) are coplanar.
\nSolution:
\n\"Inter
\n\"Inter<\/p>\n

\"Inter<\/p>\n

Question 2.
\nIf \\(\\overline{\\mathbf{a}}, \\overline{\\mathbf{b}}\\) and \\(\\overline{\\mathbf{c}}\\) non-coplanar vectors, then prove that the four points with position vectors \\(2 \\bar{a}+3 \\bar{b}-\\bar{c}\\), \\(\\overline{\\mathrm{a}}-2 \\overline{\\mathrm{b}}+3 \\overline{\\mathrm{c}}, 3 \\overline{\\mathrm{a}}+4 \\overline{\\mathrm{b}}-2 \\overline{\\mathrm{c}}\\) and \\(\\bar{a}-6 \\bar{b}+6 \\bar{c}\\) are coplanar.
\nSolution:
\nSuppose A, B, C, D are the given points.
\n\"Inter
\nThe vectors \\(\\overrightarrow{\\mathrm{AB}}, \\overrightarrow{\\mathrm{AC}}, \\overrightarrow{\\mathrm{AD}}\\) are coplanar.
\nThe given points A, B, C, D are coplanar.<\/p>\n

Question 3.
\n\\(\\overline{\\mathbf{a}}, \\overline{\\mathbf{b}}\\) and \\(\\overline{\\mathbf{c}}\\) are non-zero and non- collinear vectors and \u03b8 \u2260 0, is the angle between \\(\\overline{\\mathbf{b}}\\) and \\(\\overline{\\mathbf{c}}\\). If \\((\\overline{\\mathbf{a}} \\times \\overline{\\mathbf{b}}) \\times \\overline{\\mathbf{c}}\\) = \\(\\frac{1}{3}|\\bar{b}||\\bar{c}|\\bar{a}|\\), then find sin \u03b8.
\nSolution:
\n\"Inter<\/p>\n

Question 4.
\nFind the volume of the tetrahedron whose vertices are (1, 2, 1), (3, 2, 5), (2, -1, 0) and (-1, 0, 1).
\nSolution:
\nLet ‘O’ be the given A, B, C, D be the vertices of the ten tetrahedrons. Then
\n\"Inter
\n\"Inter<\/p>\n

Question 5.
\nShow that \\((\\bar{a}+\\bar{b}) \\cdot(\\bar{b}+\\bar{c}) \\times(\\bar{c}+\\bar{a})\\) = \\(2\\left[\\begin{array}{lll}
\n\\bar{a} & \\bar{b} & \\bar{c}
\n\\end{array}\\right]\\)
\nSolution:
\n\"Inter<\/p>\n

Question 6.
\nShow that equation of the plane passing through the points with position vectors. \\(3 \\bar{i}-5 \\bar{j}-\\overline{\\mathbf{k}},-\\overline{\\mathbf{i}}+5 \\bar{j}+7 \\overline{\\mathbf{k}}\\) and parallel to the vector \\(3 \\overline{\\mathbf{i}}-\\overline{\\mathbf{j}}+7 \\overline{\\mathbf{k}}\\) is 3x + 2y – z = 0.
\nSolution:
\nThe given plane passes through the points A, B (i.e.,) \\(3 \\bar{i}-5 \\bar{j}-\\overline{\\mathbf{k}},-\\overline{\\mathbf{i}}+5 \\bar{j}+7 \\overline{\\mathbf{k}}\\) and parallel to the vector \\(3 \\overline{\\mathbf{i}}-\\overline{\\mathbf{j}}+7 \\overline{\\mathbf{k}}\\)
\n\"Inter
\n= x(70 + 8) – y(-28 – 24) + z(4 – 30)
\n= 78x + 52y – 26z
\n= 26(3x + 2y – z)
\n\\(\\left[\\begin{array}{lll}
\n\\bar{a} & \\bar{b} & \\bar{c}
\n\\end{array}\\right]\\) = \\(\\left|\\begin{array}{rrr}
\n3 & -5 & -1 \\\\
\n-4 & 10 & 8 \\\\
\n3 & -1 & 7
\n\\end{array}\\right|\\)
\n= 3(70 + 8) + 5(-28 – 24) – 1(4 – 30)
\n= 234 – 260 + 26
\n= 0
\nEquation of the required plane is 26(3x + 2y – z) = 0
\ni.e., 3x + 2y – z = 0<\/p>\n

\"Inter<\/p>\n

Question 7.
\nProve that \\(\\overline{\\mathbf{a}} \\times[\\overline{\\mathbf{a}} \\times(\\overline{\\mathbf{a}} \\times \\overline{\\mathbf{b}})]\\) = \\((\\overline{\\mathbf{a}} \\cdot \\overline{\\mathbf{a}})(\\overline{\\mathbf{b}} \\times \\overline{\\mathbf{a}})\\)
\nSolution:
\n\"Inter<\/p>\n

Question 8.
\nIf \\(\\overline{\\mathbf{a}}, \\overline{\\mathbf{b}}, \\overline{\\mathbf{c}}\\) and \\(\\bar{d}\\) are coplanar vectors, then show that \\((\\bar{a} \\times \\bar{b}) \\times(\\bar{c} \\times \\bar{d})=0\\).
\nSolution:
\n\\(\\overline{\\mathbf{a}}, \\overline{\\mathbf{b}}, \\overline{\\mathbf{c}}\\) and \\(\\bar{d}\\) are coplanar
\n\u21d2 \\(\\overline{\\mathbf{a}} \\times \\overline{\\mathbf{b}}\\) is perpendicular to the plane \u03c0.
\nsimilarly \\(\\bar{c} \\times \\bar{d}\\) is perpendicular to the plane \u03c0.
\n\\(\\bar{a} \\times \\bar{b}\\) and \\(\\bar{c} \\times \\bar{d}\\) are parallel vectors.
\n\u21d2 \\((\\bar{a} \\times \\bar{b}) \\times(\\bar{c} \\times \\bar{d})\\) = 0.<\/p>\n

Question 9.
\nShow that \\((\\overline{\\mathrm{a}} \\times \\overline{\\mathrm{b}}) \\times(\\overline{\\mathrm{a}} \\times \\overline{\\mathrm{c}}) \\cdot \\overline{\\mathrm{d}}=[\\overline{\\mathrm{a}} \\overline{\\mathrm{b}} \\overline{\\mathrm{c}}](\\overline{\\mathrm{a}} \\cdot \\overline{\\mathrm{d}})\\).
\nSolution:
\n\"Inter<\/p>\n

Question 10.
\nShow that \\(\\bar{a} \\cdot[(\\bar{b}+\\bar{c}) \\times(\\bar{a}+\\bar{b}+\\bar{c})]=0\\)
\nSolution:
\n\"Inter<\/p>\n

Question 11.
\nFind \u03bb in order that the four points A(3, 2, 1), B(4, \u03bb, 5), C(4, 2, -2) and D(6, 5, -1) be coplanar.
\nSolution:
\n\"Inter<\/p>\n

Question 12.
\nFind the vector equation of the plane passing through the intersection of planes \\(\\bar{r} \\cdot(2 \\bar{i}+2 \\bar{j}-3 \\bar{k})=7, \\bar{r} \\cdot(2 \\bar{i}+5 \\bar{j}+3 \\bar{k})=9\\) and through the point (2, 1, 3)
\nSolution:
\n\"Inter
\n\"Inter<\/p>\n

Question 13.
\nFind the equation of the plane passing through (a, b, c) and parallel to the plane \\(\\bar{r} \\cdot(\\bar{i}+\\bar{i}+\\bar{k})=2\\).
\nSolution:
\nGiven equation plane is \\(\\bar{r} \\cdot(\\bar{i}+\\bar{i}+\\bar{k})=2\\)
\nLet \\(\\bar{r}=x \\bar{i}+y \\bar{j}+z \\bar{k}\\)
\n\u2234 \\(\\bar{r} \\cdot(\\bar{i}+\\bar{i}+\\bar{k})=2\\)
\n\\((x \\bar{i}+y \\bar{j}+z \\bar{k}) \\cdot(i+j+k)=2\\)
\nx + y + z = 2
\nRequired plane equation is x + y + z = k …….(1)
\nEquation (1) passes through (a, b, c)
\n\u2234 a + b + c = k
\nSubstitute ‘k’ in equation (1)
\n\u2234 x + y + z = a + b + c<\/p>\n

\"Inter<\/p>\n

Question 14.
\nFind the shortest distance between the lines \\(\\bar{r}=6 \\bar{i}+2 \\bar{j}+2 \\bar{k}+\\lambda, \\bar{i}-2 \\bar{j}+2 \\bar{k}\\) and \\(\\bar{r}=-4 \\bar{j}-\\bar{k}+\\mu=3 \\bar{j}-2 \\bar{j}-2 \\bar{k}\\).
\nSolution:
\nThe first line passes through point A(6, 2, 2) and is parallel to the vector b = i – 2j + 2k.
\nSecond line passes through the point C(-4, 0, -1) and is parallel to the vector d = 3i – 2j – 2k
\n\"Inter
\n\"Inter<\/p>\n

Question 15.
\nFind the equation of the plane passing through the line of intersection of the planes \\(\\bar{r} \\cdot(\\bar{i}+\\bar{j}+\\bar{k})=1\\) and \\(\\bar{r} \\cdot(2 \\bar{i}+3 \\bar{i}-\\bar{k})+4=0\\) and parallel to X-axis.
\nSolution:
\nGiven the equation of planes are
\n\"Inter
\nSince it is parallel to X-axis.<\/p>\n

Question 16.
\nProve that the four points \\(4 \\bar{i}+5 \\bar{j}+\\bar{k}\\), \\(-(\\overline{\\mathbf{j}}+\\overline{\\mathbf{k}}), 3 \\overline{\\mathbf{i}}+9 \\overline{\\mathbf{j}}+4 \\overline{\\mathbf{k}}\\) and \\(-4 \\bar{i}+4 \\bar{j}+4 \\bar{k}\\) are coplanar.
\nSolution:
\nLet \u2018O’ be the origin. A, B, C, D be the given points. Then
\n\"Inter
\n\"Inter<\/p>\n

Question 17.
\nIf \\(\\bar{a}, \\bar{b}, \\bar{c}\\) are non – copianar, then show that the vectors \\(\\overline{\\mathbf{a}}-\\overline{\\mathbf{b}}, \\overline{\\mathbf{b}}+\\overline{\\mathbf{c}}\\), \\(\\overline{\\mathbf{c}}+\\overline{\\mathbf{a}}\\) are coplanar.
\nSolution:
\n\"Inter<\/p>\n

\"Inter<\/p>\n

Question 18.
\nIf \\(\\overline{\\mathbf{a}}, \\overline{\\mathbf{b}}, \\overline{\\mathbf{c}}\\) are the position vectors of the points A, B and C respectively, then prove that the vector \\(\\overline{\\mathbf{a}} \\times \\overline{\\mathbf{b}}+\\overline{\\mathbf{b}} \\times \\overline{\\mathbf{c}}+\\overline{\\mathbf{c}} \\times \\overline{\\mathbf{a}}\\) is perpendicular to the plane of \u2206ABC.
\nSolution:
\n\"Inter
\n\"Inter<\/p>\n

III.<\/span><\/p>\n

Question 1.
\nShow that \\((\\bar{a} \\times(\\bar{b} \\times \\bar{c}) \\times \\bar{c})=(\\bar{a} \\cdot \\bar{c})(\\bar{b} \\times \\bar{c})\\) and \\((\\overline{\\mathbf{a}} \\times \\overline{\\mathbf{b}}) \\cdot(\\overline{\\mathbf{a}} \\times \\overline{\\mathbf{c}})+(\\overline{\\mathbf{a}} \\cdot \\overline{\\mathbf{b}})(\\overline{\\mathbf{a}} \\cdot \\overline{\\mathbf{c}})\\) = \\((\\overline{\\mathbf{a}} \\cdot \\overline{\\mathbf{a}})(\\overline{\\mathbf{b}} \\cdot \\overline{\\mathbf{c}})\\)
\nSolution:
\n\"Inter<\/p>\n

Question 2.
\nIf A = (1, -2, -1), B = (4, 0, -3), C = (1, 2, -1) and D = (2, -4, -5), find the distance between AB and CD.
\nSolution:
\n\"Inter
\n\"Inter<\/p>\n

Question 3.
\nIf \\(\\overline{\\mathbf{a}}=\\overline{\\mathbf{i}}-2 \\overline{\\mathbf{j}}+\\overline{\\mathbf{k}}, \\overline{\\mathbf{b}}=2 \\overline{\\mathbf{i}}+\\overline{\\mathbf{j}}+\\overline{\\mathbf{k}}\\) and \\(\\overline{\\mathbf{c}}=\\overline{\\mathbf{i}}+2 \\overline{\\mathbf{j}}-\\overline{\\mathbf{k}}\\), find \\(\\bar{a} \\times(\\bar{b} \\times \\bar{c})\\) and \\(|(\\overline{\\mathbf{a}} \\times \\overline{\\mathbf{b}}) \\times \\overline{\\mathbf{c}}|\\).
\nSolution:
\n\"Inter
\n\"Inter<\/p>\n

\"Inter<\/p>\n

Question 4.
\nIf \\(\\overline{\\mathbf{a}}=\\overline{\\mathbf{i}}-2 \\overline{\\mathbf{j}}-3 \\overline{\\mathbf{k}}, \\overline{\\mathbf{b}}=2 \\overline{\\mathbf{i}}+\\overline{\\mathbf{j}}-\\overline{\\mathbf{k}}\\) and \\(\\bar{c}=\\bar{i}+3 \\bar{j}-2 \\bar{k}\\), verift that \\(\\overline{\\mathbf{a}} \\times(\\overline{\\mathbf{b}} \\times \\overline{\\mathbf{c}}) \\neq(\\overline{\\mathbf{a}} \\times \\overline{\\mathbf{b}}) \\times \\overline{\\mathbf{c}}\\)
\nSolution:
\n\"Inter
\nFrom (1) and (2), we get
\n\\(\\overline{\\mathbf{a}} \\times(\\overline{\\mathbf{b}} \\times \\overline{\\mathbf{c}}) \\neq(\\overline{\\mathbf{a}} \\times \\overline{\\mathbf{b}}) \\times \\overline{\\mathbf{c}}\\)
\ni.e., vector multiplication is not associative.<\/p>\n

Question 5.
\nIf \\(\\overline{\\mathbf{a}}=2 \\overline{\\mathbf{i}}+\\overline{\\mathbf{j}}-3 \\overline{\\mathbf{k}}, \\overline{\\mathbf{b}}=\\overline{\\mathbf{i}}-\\mathbf{2} \\mathbf{j}+\\overline{\\mathbf{k}}\\), \\(\\bar{c}=-\\bar{i}+\\bar{j}-4 \\bar{k}\\) and \\(\\overline{\\mathbf{d}}=\\overline{\\mathbf{i}}+\\overline{\\mathbf{j}}+\\overline{\\mathbf{k}}\\), then compute \\(|(\\overline{\\mathrm{a}} \\times \\overline{\\mathrm{b}}) \\times(\\overline{\\mathrm{c}} \\times \\overline{\\mathrm{d}})|\\)
\nSolution:
\n\"Inter<\/p>\n

Question 6.
\nIf A = (1, a, a2<\/sup>), B = (1, b, b2<\/sup>) and C = (1, c, c2<\/sup>) are non-coplanar vectors and \\(\\left|\\begin{array}{lll}
\na & a^{2} & 1+a^{3} \\\\
\nb & b^{2} & 1+b^{3} \\\\
\nc & c^{2} & 1+c^{3}
\n\\end{array}\\right|\\) = 0, then show that abc + 1 = 0
\nSolution:
\n\\(\\bar{A}, \\bar{B}, \\bar{C}\\) are non-coplanar vectors.
\n\u2206 = \\(\\left|\\begin{array}{lll}
\n1 & a & a^{2} \\\\
\n1 & b & b^{2} \\\\
\n1 & c & c^{2}
\n\\end{array}\\right|\\) \u2260 0 ………..(1)
\n\"Inter
\n\u2206 + (abc) \u2206 = 0; \u2206(1 + abc) = 0
\n\u2206 \u2260 0 \u21d2 1 + abc = 0<\/p>\n

\"Inter<\/p>\n

Question 7.
\nIf \\(\\overline{\\mathbf{a}}, \\overline{\\mathbf{b}}, \\overline{\\mathbf{c}}\\) are non-zero vectors, then \\(|(\\overline{\\mathbf{a}} \\times \\mathbf{b} \\cdot \\overline{\\mathbf{c}})|=|\\overline{\\mathbf{a}}||\\mathbf{b}||\\overline{\\mathbf{c}}|\\) \\(\\Leftrightarrow \\overline{\\mathbf{a}} \\cdot \\overline{\\mathbf{b}}=\\overline{\\mathbf{b}} \\cdot \\overline{\\mathbf{c}}=\\overline{\\mathbf{c}} \\cdot \\overline{\\mathbf{a}}=\\mathbf{0}\\)
\nSolution:
\n\"Inter<\/p>\n

Question 8.
\nIf \\(\\overline{\\mathbf{a}}=\\overline{\\mathbf{i}}-\\mathbf{2} \\overline{\\mathbf{j}}+3 \\overline{\\mathbf{k}}, \\quad \\mathbf{b}=2 \\overline{\\mathbf{i}}+\\overline{\\mathbf{j}}+\\overline{\\mathbf{k}}\\), \\(\\bar{c}=\\overline{\\mathbf{i}}+\\overline{\\mathbf{j}}+2 \\overline{\\mathbf{k}}\\) then find \\(|(\\bar{a} \\times \\bar{b}) \\times \\bar{c}|\\) and \\(|\\overline{\\mathbf{a}} \\times(\\mathbf{b} \\times \\mathbf{c})|\\).
\nSolution:
\n\"Inter
\n\"Inter<\/p>\n

Question 9.
\nIf \\(|\\bar{a}|=1,|\\bar{b}|=1,|\\bar{c}|=2\\) and \\(\\bar{a} \\times(a \\times \\bar{c})+\\bar{b}=0\\) then find the angle between \\(\\bar{a}\\) and \\(\\bar{c}\\).
\nSolution:
\n\"Inter
\n\"Inter<\/p>\n

Question 10.
\nLet \\(\\overline{\\mathbf{a}}=\\overline{\\mathbf{i}}-\\overline{\\mathbf{k}}, \\quad \\overline{\\mathbf{b}}=\\mathbf{x} \\overline{\\mathbf{i}}+\\overline{\\mathbf{j}}+(\\mathbf{1}-\\mathbf{x}) \\overline{\\mathbf{k}}\\) and \\(\\bar{c}=y \\bar{i}+x \\bar{j}+(1+x-y) \\bar{k}\\), prove that the scalar triple product \\(\\left[\\begin{array}{lll}
\n\\bar{a} & \\bar{b} & \\bar{c}
\n\\end{array}\\right]\\) is independent of both x and y.
\nSolution:
\n\"Inter<\/p>\n

Question 11.
\nLet \\(\\overline{\\mathbf{b}}=\\mathbf{2} \\overline{\\mathbf{i}}+\\overline{\\mathbf{j}}-\\overline{\\mathbf{k}}, \\overline{\\mathbf{c}}=\\overline{\\mathbf{i}}+\\mathbf{3} \\overline{\\mathbf{k}}\\). If \\(\\overline{\\mathrm{a}}\\) is a unit vector then find the maximum value of \\(\\left[\\begin{array}{lll}
\n\\overline{\\mathbf{a}} & \\overline{\\mathbf{b}} & \\bar{c}
\n\\end{array}\\right]\\).
\nSolution:
\nLet \\(\\bar{a}=x \\bar{i}+y \\bar{j}+z \\bar{k}\\) and x2<\/sup> + y2<\/sup> + z2<\/sup> = 1
\n\u2235 \\(\\overline{\\mathrm{a}}\\) unit vector
\n\"Inter<\/p>\n

\"Inter<\/p>\n

Question 12.
\nLet \\(\\overline{\\mathrm{a}}=\\overline{\\mathrm{i}}-\\overline{\\mathrm{i}}, \\overline{\\mathrm{b}}=\\overline{\\mathrm{i}}-\\overline{\\mathrm{k}}, \\overline{\\mathrm{c}}=\\overline{\\mathrm{k}}-\\overline{\\mathrm{i}}\\) Find unit vector \\(\\bar{d}\\) such that \\(\\overline{\\mathrm{a}} \\cdot \\overline{\\mathrm{d}}=0=[\\bar{b} \\overline{\\mathrm{c}} \\overline{\\mathrm{d}}]\\)
\nSolution:
\n\"Inter
\n\"Inter<\/p>\n","protected":false},"excerpt":{"rendered":"

Practicing the Intermediate 1st Year Maths 1A Textbook Solutions Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(c) will help students to clear their doubts quickly. Intermediate 1st Year Maths 1A Products of Vectors Solutions Exercise 5(c) I. Question 1. Compute Solution: Question 2. If , , then compute . Solution: Question 3. … Read more<\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[14],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/21879"}],"collection":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/comments?post=21879"}],"version-history":[{"count":1,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/21879\/revisions"}],"predecessor-version":[{"id":21880,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/21879\/revisions\/21880"}],"wp:attachment":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/media?parent=21879"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/categories?post=21879"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/tags?post=21879"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}