Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c)

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 \([\overline{\mathbf{i}}-\overline{\mathbf{j}} \overline{\mathbf{j}}-\overline{\mathbf{k}} \overline{\mathbf{k}}-\overline{\mathbf{i}}]\)
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q1

Question 2.
If \(\overline{\mathbf{a}}=\overline{\mathbf{i}}-2 \overline{\mathbf{j}}-3 \overline{\mathbf{k}}, \overline{\mathbf{b}}=2 \overline{\mathbf{i}}+\overline{\mathbf{j}}-\overline{\mathbf{k}}\), \(\bar{c}=\bar{i}+3 \bar{j}-2 \bar{k}\), then compute \(\overline{\mathbf{a}} \cdot(\overline{\mathbf{b}} \times \overline{\mathbf{c}})\).
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q2

Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c)

Question 3.
If \(\bar{a}\) = (1, -1, -6), \(\bar{b}\) = (1, -3, 4) and \(\bar{c}\) = (2, -5, 3), then compute the following
(i) \(\overline{\mathbf{a}} \cdot(\bar{b} \times \bar{c})\)
(ii) \(\overline{\mathbf{a}} \times(\overline{\mathbf{b}} \times \overline{\mathbf{c}})\)
(iii) \((\overline{\mathbf{a}} \times \overline{\mathbf{b}}) \times \overline{\mathbf{c}}\)
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q3
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q3.1

Question 4.
Simplify the following.
(i) \((\bar{i}-2 \bar{j}+3 \bar{k}) \times(2 i+j-\bar{k}) \cdot(\bar{j}+\bar{k})\)
(ii) \((2 \bar{i}-3 \bar{j}+\bar{k}) \cdot(\bar{i}-\bar{j}+2 \bar{k}) \cdot(2 \bar{i}+\bar{j}+\bar{k})\)
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q4

Question 5.
Find the volume of the parallelopiped having coterminous edges.
\(\overline{\mathbf{i}}+\overline{\mathbf{j}}+\overline{\mathbf{k}}, \overline{\mathbf{i}}-\overline{\mathbf{j}}\) and \(\overline{\mathbf{i}}+\mathbf{2} \overline{\mathbf{j}}-\overline{\mathbf{k}}\)
Solution:
Let \(\overline{\mathrm{a}}=\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{b}}=\overline{\mathrm{i}}-\overline{\mathrm{j}}\) and \(\overline{\mathrm{c}}=\overline{\mathrm{i}}+2 \overline{\mathrm{j}}-\overline{\mathrm{k}}\)
Volume of the parallelopiped = \([(\bar{a} \bar{b} \bar{c})]\)
= \(\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & -1 & 0 \\
1 & 2 & -1
\end{array}\right|\)
= 1(1 – 0) – 1(-1 – 0) + 1(2 + 1)
= 1 + 1 + 3
= 5 cubic units.

Question 6.
Find 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.
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q6

Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c)

Question 7.
For 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.
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q7

Question 8.
Determine λ, 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.
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q8

Question 9.
Find the volume of the tetrahedron having the edges \(\overline{\mathbf{i}}+\overline{\mathbf{j}}+\overline{\mathbf{k}}, \quad \mathbf{i}-\overline{\mathbf{j}}\) and \(\bar{i}+2 \bar{j}+\bar{k}\).
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q9

Question 10.
Let \(\bar{a}\), \(\bar{b}\) 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 \(\left[\begin{array}{lll}
\bar{\alpha} & \bar{\beta} & \bar{\gamma}
\end{array}\right]\).
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q10

Question 11.
Let \(\bar{a}\), \(\bar{b}\) 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 λ.
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q11
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q11.1

Question 12.
Let \(\bar{a}\), \(\bar{b}\) and \(\bar{c}\) be non-coplanar vectors, if \(\left[\begin{array}{lll}
\bar{a}+2 \bar{b} & 2 \bar{b}+\bar{c} & 5 \bar{c}+\bar{a}
\end{array}\right]\) = \(\lambda\left[\begin{array}{lll}
\bar{a} & \bar{b} & \bar{c}
\end{array}\right]\), then find λ.
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q12

Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c)

Question 13.
If \(\bar{a}\), \(\bar{b}\) and \(\bar{c}\) are non-coplanar vectors, then find the value of \(\frac{(\bar{a}+2 \bar{b}-\bar{c}) \cdot[(\bar{a}-\bar{b}) \times(\bar{a}-\bar{b}-\bar{c})]}{[\bar{a} \bar{b} \bar{c}]}\)
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q13

Question 14.
If \(\bar{a}\), \(\bar{b}\) and \(\bar{c}\) are mutually perpendicular unit vectors, then find the value of \(\left[\begin{array}{lll}
\bar{a} & \bar{b} & \bar{c}
\end{array}\right]^{2}\).
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q14

Question 15.
\(\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}\) 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 \(|[\bar{a} \bar{b} \bar{c}]|\).
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q15

Question 16.
If \(\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}\) are unit coplanar vectors, then find \(\left[\begin{array}{lll}
2 \bar{a}-\bar{b} & 2 \bar{b}-\bar{c} & 2 \bar{c}-\bar{a}
\end{array}\right]\)
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) I Q16

II.

Question 1.
If \(\left[\begin{array}{lll}
\bar{b} & \bar{c} & \bar{d}
\end{array}\right]+\left[\begin{array}{lll}
\bar{c} & \bar{a} & \bar{d}
\end{array}\right]+\left[\begin{array}{lll}
\bar{a} & \bar{b} & \bar{d}
\end{array}\right]\) = \(\left[\begin{array}{lll}
\overline{\mathbf{a}} & \overline{\mathbf{b}} & \overline{\mathbf{c}}
\end{array}\right]\) then show that the points with position vectors \(\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}\) and \(\bar{d}\) are coplanar.
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q1
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q1.1

Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c)

Question 2.
If \(\overline{\mathbf{a}}, \overline{\mathbf{b}}\) 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.
Solution:
Suppose A, B, C, D are the given points.
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q2
The vectors \(\overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AC}}, \overrightarrow{\mathrm{AD}}\) are coplanar.
The given points A, B, C, D are coplanar.

Question 3.
\(\overline{\mathbf{a}}, \overline{\mathbf{b}}\) and \(\overline{\mathbf{c}}\) are non-zero and non- collinear vectors and θ ≠ 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 θ.
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q3

Question 4.
Find the volume of the tetrahedron whose vertices are (1, 2, 1), (3, 2, 5), (2, -1, 0) and (-1, 0, 1).
Solution:
Let ‘O’ be the given A, B, C, D be the vertices of the ten tetrahedrons. Then
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q4
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q4.1

Question 5.
Show that \((\bar{a}+\bar{b}) \cdot(\bar{b}+\bar{c}) \times(\bar{c}+\bar{a})\) = \(2\left[\begin{array}{lll}
\bar{a} & \bar{b} & \bar{c}
\end{array}\right]\)
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q5

Question 6.
Show 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.
Solution:
The 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 \(3 \overline{\mathbf{i}}-\overline{\mathbf{j}}+7 \overline{\mathbf{k}}\)
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q6
= x(70 + 8) – y(-28 – 24) + z(4 – 30)
= 78x + 52y – 26z
= 26(3x + 2y – z)
\(\left[\begin{array}{lll}
\bar{a} & \bar{b} & \bar{c}
\end{array}\right]\) = \(\left|\begin{array}{rrr}
3 & -5 & -1 \\
-4 & 10 & 8 \\
3 & -1 & 7
\end{array}\right|\)
= 3(70 + 8) + 5(-28 – 24) – 1(4 – 30)
= 234 – 260 + 26
= 0
Equation of the required plane is 26(3x + 2y – z) = 0
i.e., 3x + 2y – z = 0

Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c)

Question 7.
Prove that \(\overline{\mathbf{a}} \times[\overline{\mathbf{a}} \times(\overline{\mathbf{a}} \times \overline{\mathbf{b}})]\) = \((\overline{\mathbf{a}} \cdot \overline{\mathbf{a}})(\overline{\mathbf{b}} \times \overline{\mathbf{a}})\)
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q7

Question 8.
If \(\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}\) and \(\bar{d}\) are coplanar vectors, then show that \((\bar{a} \times \bar{b}) \times(\bar{c} \times \bar{d})=0\).
Solution:
\(\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}\) and \(\bar{d}\) are coplanar
⇒ \(\overline{\mathbf{a}} \times \overline{\mathbf{b}}\) is perpendicular to the plane π.
similarly \(\bar{c} \times \bar{d}\) is perpendicular to the plane π.
\(\bar{a} \times \bar{b}\) and \(\bar{c} \times \bar{d}\) are parallel vectors.
⇒ \((\bar{a} \times \bar{b}) \times(\bar{c} \times \bar{d})\) = 0.

Question 9.
Show that \((\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times(\overline{\mathrm{a}} \times \overline{\mathrm{c}}) \cdot \overline{\mathrm{d}}=[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}](\overline{\mathrm{a}} \cdot \overline{\mathrm{d}})\).
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q9

Question 10.
Show that \(\bar{a} \cdot[(\bar{b}+\bar{c}) \times(\bar{a}+\bar{b}+\bar{c})]=0\)
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q10

Question 11.
Find λ in order that the four points A(3, 2, 1), B(4, λ, 5), C(4, 2, -2) and D(6, 5, -1) be coplanar.
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q11

Question 12.
Find 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)
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q12
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q12.1

Question 13.
Find the equation of the plane passing through (a, b, c) and parallel to the plane \(\bar{r} \cdot(\bar{i}+\bar{i}+\bar{k})=2\).
Solution:
Given equation plane is \(\bar{r} \cdot(\bar{i}+\bar{i}+\bar{k})=2\)
Let \(\bar{r}=x \bar{i}+y \bar{j}+z \bar{k}\)
∴ \(\bar{r} \cdot(\bar{i}+\bar{i}+\bar{k})=2\)
\((x \bar{i}+y \bar{j}+z \bar{k}) \cdot(i+j+k)=2\)
x + y + z = 2
Required plane equation is x + y + z = k …….(1)
Equation (1) passes through (a, b, c)
∴ a + b + c = k
Substitute ‘k’ in equation (1)
∴ x + y + z = a + b + c

Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c)

Question 14.
Find 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 \(\bar{r}=-4 \bar{j}-\bar{k}+\mu=3 \bar{j}-2 \bar{j}-2 \bar{k}\).
Solution:
The first line passes through point A(6, 2, 2) and is parallel to the vector b = i – 2j + 2k.
Second line passes through the point C(-4, 0, -1) and is parallel to the vector d = 3i – 2j – 2k
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q14
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q14.1

Question 15.
Find 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.
Solution:
Given the equation of planes are
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q15
Since it is parallel to X-axis.

Question 16.
Prove 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.
Solution:
Let ‘O’ be the origin. A, B, C, D be the given points. Then
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q16
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q16.1

Question 17.
If \(\bar{a}, \bar{b}, \bar{c}\) 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.
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q17

Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c)

Question 18.
If \(\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}\) 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 ∆ABC.
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q18
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) II Q18.1

III.

Question 1.
Show that \((\bar{a} \times(\bar{b} \times \bar{c}) \times \bar{c})=(\bar{a} \cdot \bar{c})(\bar{b} \times \bar{c})\) and \((\overline{\mathbf{a}} \times \overline{\mathbf{b}}) \cdot(\overline{\mathbf{a}} \times \overline{\mathbf{c}})+(\overline{\mathbf{a}} \cdot \overline{\mathbf{b}})(\overline{\mathbf{a}} \cdot \overline{\mathbf{c}})\) = \((\overline{\mathbf{a}} \cdot \overline{\mathbf{a}})(\overline{\mathbf{b}} \cdot \overline{\mathbf{c}})\)
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q1

Question 2.
If A = (1, -2, -1), B = (4, 0, -3), C = (1, 2, -1) and D = (2, -4, -5), find the distance between AB and CD.
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q2
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q2.1

Question 3.
If \(\overline{\mathbf{a}}=\overline{\mathbf{i}}-2 \overline{\mathbf{j}}+\overline{\mathbf{k}}, \overline{\mathbf{b}}=2 \overline{\mathbf{i}}+\overline{\mathbf{j}}+\overline{\mathbf{k}}\) and \(\overline{\mathbf{c}}=\overline{\mathbf{i}}+2 \overline{\mathbf{j}}-\overline{\mathbf{k}}\), find \(\bar{a} \times(\bar{b} \times \bar{c})\) and \(|(\overline{\mathbf{a}} \times \overline{\mathbf{b}}) \times \overline{\mathbf{c}}|\).
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q3
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q3.1

Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c)

Question 4.
If \(\overline{\mathbf{a}}=\overline{\mathbf{i}}-2 \overline{\mathbf{j}}-3 \overline{\mathbf{k}}, \overline{\mathbf{b}}=2 \overline{\mathbf{i}}+\overline{\mathbf{j}}-\overline{\mathbf{k}}\) and \(\bar{c}=\bar{i}+3 \bar{j}-2 \bar{k}\), verift that \(\overline{\mathbf{a}} \times(\overline{\mathbf{b}} \times \overline{\mathbf{c}}) \neq(\overline{\mathbf{a}} \times \overline{\mathbf{b}}) \times \overline{\mathbf{c}}\)
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q4
From (1) and (2), we get
\(\overline{\mathbf{a}} \times(\overline{\mathbf{b}} \times \overline{\mathbf{c}}) \neq(\overline{\mathbf{a}} \times \overline{\mathbf{b}}) \times \overline{\mathbf{c}}\)
i.e., vector multiplication is not associative.

Question 5.
If \(\overline{\mathbf{a}}=2 \overline{\mathbf{i}}+\overline{\mathbf{j}}-3 \overline{\mathbf{k}}, \overline{\mathbf{b}}=\overline{\mathbf{i}}-\mathbf{2} \mathbf{j}+\overline{\mathbf{k}}\), \(\bar{c}=-\bar{i}+\bar{j}-4 \bar{k}\) and \(\overline{\mathbf{d}}=\overline{\mathbf{i}}+\overline{\mathbf{j}}+\overline{\mathbf{k}}\), then compute \(|(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times(\overline{\mathrm{c}} \times \overline{\mathrm{d}})|\)
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q5

Question 6.
If A = (1, a, a2), B = (1, b, b2) and C = (1, c, c2) are non-coplanar vectors and \(\left|\begin{array}{lll}
a & a^{2} & 1+a^{3} \\
b & b^{2} & 1+b^{3} \\
c & c^{2} & 1+c^{3}
\end{array}\right|\) = 0, then show that abc + 1 = 0
Solution:
\(\bar{A}, \bar{B}, \bar{C}\) are non-coplanar vectors.
∆ = \(\left|\begin{array}{lll}
1 & a & a^{2} \\
1 & b & b^{2} \\
1 & c & c^{2}
\end{array}\right|\) ≠ 0 ………..(1)
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q6
∆ + (abc) ∆ = 0; ∆(1 + abc) = 0
∆ ≠ 0 ⇒ 1 + abc = 0

Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c)

Question 7.
If \(\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}\) are non-zero vectors, then \(|(\overline{\mathbf{a}} \times \mathbf{b} \cdot \overline{\mathbf{c}})|=|\overline{\mathbf{a}}||\mathbf{b}||\overline{\mathbf{c}}|\) \(\Leftrightarrow \overline{\mathbf{a}} \cdot \overline{\mathbf{b}}=\overline{\mathbf{b}} \cdot \overline{\mathbf{c}}=\overline{\mathbf{c}} \cdot \overline{\mathbf{a}}=\mathbf{0}\)
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q7

Question 8.
If \(\overline{\mathbf{a}}=\overline{\mathbf{i}}-\mathbf{2} \overline{\mathbf{j}}+3 \overline{\mathbf{k}}, \quad \mathbf{b}=2 \overline{\mathbf{i}}+\overline{\mathbf{j}}+\overline{\mathbf{k}}\), \(\bar{c}=\overline{\mathbf{i}}+\overline{\mathbf{j}}+2 \overline{\mathbf{k}}\) then find \(|(\bar{a} \times \bar{b}) \times \bar{c}|\) and \(|\overline{\mathbf{a}} \times(\mathbf{b} \times \mathbf{c})|\).
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q8
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q8.1

Question 9.
If \(|\bar{a}|=1,|\bar{b}|=1,|\bar{c}|=2\) and \(\bar{a} \times(a \times \bar{c})+\bar{b}=0\) then find the angle between \(\bar{a}\) and \(\bar{c}\).
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q9
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q9.1

Question 10.
Let \(\overline{\mathbf{a}}=\overline{\mathbf{i}}-\overline{\mathbf{k}}, \quad \overline{\mathbf{b}}=\mathbf{x} \overline{\mathbf{i}}+\overline{\mathbf{j}}+(\mathbf{1}-\mathbf{x}) \overline{\mathbf{k}}\) and \(\bar{c}=y \bar{i}+x \bar{j}+(1+x-y) \bar{k}\), prove that the scalar triple product \(\left[\begin{array}{lll}
\bar{a} & \bar{b} & \bar{c}
\end{array}\right]\) is independent of both x and y.
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q10

Question 11.
Let \(\overline{\mathbf{b}}=\mathbf{2} \overline{\mathbf{i}}+\overline{\mathbf{j}}-\overline{\mathbf{k}}, \overline{\mathbf{c}}=\overline{\mathbf{i}}+\mathbf{3} \overline{\mathbf{k}}\). If \(\overline{\mathrm{a}}\) is a unit vector then find the maximum value of \(\left[\begin{array}{lll}
\overline{\mathbf{a}} & \overline{\mathbf{b}} & \bar{c}
\end{array}\right]\).
Solution:
Let \(\bar{a}=x \bar{i}+y \bar{j}+z \bar{k}\) and x2 + y2 + z2 = 1
∵ \(\overline{\mathrm{a}}\) unit vector
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q11

Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c)

Question 12.
Let \(\overline{\mathrm{a}}=\overline{\mathrm{i}}-\overline{\mathrm{i}}, \overline{\mathrm{b}}=\overline{\mathrm{i}}-\overline{\mathrm{k}}, \overline{\mathrm{c}}=\overline{\mathrm{k}}-\overline{\mathrm{i}}\) Find unit vector \(\bar{d}\) such that \(\overline{\mathrm{a}} \cdot \overline{\mathrm{d}}=0=[\bar{b} \overline{\mathrm{c}} \overline{\mathrm{d}}]\)
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q12
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c) III Q12.1