Practicing the Intermediate 1st Year Maths 1A Textbook Solutions Inter 1st Year Maths 1A Addition of Vectors Solutions Exercise 4(b) will help students to clear their doubts quickly.

## Intermediate 1st Year Maths 1A Addition of Vectors Solutions Exercise 4(b)

I.

Question 1.

Find the vector equation of the line passing through the point \(2 \bar{i}+3 \bar{j}+\bar{k}\) and parallel to the vector \(4 \bar{i}-2 \bar{j}+3 \bar{k}\).

Solution:

Question 2.

OABC is a parallelogram. If \(\overline{O A}=\bar{a}\) and \(\overline{O C}=\bar{c}\), find the vector equation of the side BC.

Solution:

Question 3.

If \(\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}\) are the position vectors of the vertices A, B and C respectively of ∆ABC, theind the vector equation of the median through the vertex A.

Solution:

Question 4.

Find the vertor equation of the line joining the points \(2 \bar{i}+\bar{j}+3 \bar{k}\) and \(-4 \bar{i}+3 \bar{j}-\bar{k}\).

Solution:

Question 5.

Find the vector equation of the plane passing through the points \(\overline{\mathbf{i}}-2 \overline{\mathbf{j}}+5 \overline{\mathbf{k}},-5 \overline{\mathbf{j}}-\overline{\mathbf{k}} \text { and }-3 \overline{\mathbf{i}}+5 \overline{\mathbf{j}}\).

Solution:

Question 6.

Find the vector equation of the plane through the points (0, 0, 0), (0, 5, 0) and (2, 0, 1).

Solution:

II.

Question 1.

If \(\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}\) are noncoplanar find the point of intersection of the line passing through the points \(2 \bar{a}+3 \bar{b}-\bar{c}\), \(3 \bar{a}+4 \bar{b}-2 \bar{c}\) with the line joining the points \(\bar{a}-2 \bar{b}+3 \bar{c}, \bar{a}-6 \bar{b}+6 \bar{c}\).

Solution:

Question 2.

ABCD is a trapezium in which AB and CD are parallel. Prove by vector methods, that the mid points of the sides AB, CD and the intersection of the diagonals are collinear.

Solution:

⇒ M, P, N are collinear

Hence the midpoints of parallel sides of a trapezium and the point of intersection of the diagonals are collinear.

Question 3.

In a quadrilateral ABCD, if the midpoints of one pair of opposite sides and the point of intersection of the diagonals are collinear, using vector methods, prove that the quadrilateral ABCD is a trapezium.

Solution:

III.

Question 1.

Find the vector equation of the plane which passes through the points \(2 \bar{i}+4 \bar{j}+2 \bar{k}, 2 \bar{i}+3 \bar{j}+5 \bar{k}\) and parallel to the vector \(3 \overline{\mathbf{i}}-2 \overline{\mathbf{j}}+\overline{\mathbf{k}}\). Also find the point where this plane meets the line joining the points \(2 \overline{\mathbf{i}}+\overline{\mathbf{j}}+3 \overline{\mathbf{k}}\) and \(4 \bar{i}-2 \bar{j}+3 \bar{k}\).

Solution:

Question 2.

Find the vector equation of the plane passing through points \(4 \overline{\mathbf{i}}-3 \overline{\mathbf{j}}-\overline{\mathbf{k}}\), \(3 \overline{\mathbf{i}}+7 \overline{\mathbf{j}}-10 \overline{\mathbf{k}}\) and \(2 \bar{i}+5 \bar{j}-7 \bar{k}\), and show that the point \(\overline{\mathbf{i}}+2 \bar{j}-3 \overline{\mathbf{k}}\) lies in the plane.

Solution: