Practicing the Intermediate 1st Year Maths 1A Textbook Solutions Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(b) will help students to clear their doubts quickly.

## Intermediate 1st Year Maths 1A Products of Vectors Solutions Exercise 5(b)

I.

Question 1.
If $$|\overline{\mathbf{p}}|=2,|\overline{\mathbf{q}}|=3$$ and $$(\bar{p}, \bar{q})=\frac{\pi}{6}$$, then find $$|\bar{p} \times \bar{q}|^{2}$$.
Solution:

Question 2.
If $$\overline{\mathbf{a}}=2 \overline{\mathbf{i}}-\overline{\mathbf{j}}+\overline{\mathbf{k}}$$ and $$\overline{\mathbf{b}}=\overline{\mathbf{i}}-3 \overline{\mathbf{j}}-5 \overline{\mathbf{k}}$$, then find $$|\overline{\mathbf{a}} \times \overline{\mathbf{b}}|$$.
Solution:

Question 3.
If $$\bar{a}=2 \bar{i}-3 \bar{j}+\overline{\mathbf{k}}$$ and $$\bar{b}=\bar{i}+4 \bar{j}-2 \bar{k}$$, then find $$(\overline{\mathbf{a}}+\overline{\mathbf{b}}) \times(\overline{\mathbf{a}}-\overline{\mathbf{b}})$$.
Solution:

Question 4.
If $$4 \bar{i}+\frac{2 p}{3} \bar{j}+p \bar{k}$$ is parallel to the vector $$\overline{\mathbf{i}}+2 \overline{\mathbf{j}}+3 \overline{\mathbf{k}}$$, find p.
Solution:

Question 5.
Compute $$\overline{\mathbf{a}} \times(\overline{\mathbf{b}}+\overline{\mathbf{c}})+\overline{\mathbf{b}} \times(\overline{\mathbf{c}}+\overline{\mathbf{a}})+\overline{\mathbf{c}} \times(\overline{\mathbf{a}}+\overline{\mathbf{b}})$$
Solution:

Question 6.
If $$\overline{\mathbf{p}}=\mathbf{x} \overline{\mathbf{i}}+\mathbf{y} \overline{\mathbf{j}}+\mathbf{z} \overline{\mathbf{k}}$$, find the value of $$|\overline{\boldsymbol{p}} \times \overline{\mathbf{k}}|^{2}$$
Solution:

Question 7.
Compute $$2 \bar{j} \times(3 \bar{i}-4 \bar{k})+(\bar{i}+2 \hat{j}) \times \bar{k}$$.
Solution:

Question 8.
Find a unit vector perpendicular to both $$\overline{\mathbf{i}}+\overline{\mathbf{j}}+\overline{\mathbf{k}}$$ and $$2 \bar{i}+\bar{j}+3 \bar{k}$$.
Solution:

Question 9.
If θ is the angle between the vectors $$\overline{\mathbf{i}}+\overline{\mathbf{j}}$$ and $$\overline{\mathbf{j}}+\overline{\mathbf{k}}$$, then find sin ?.
Solution:

Question 10.
Find the area of the parallelogram having $$\bar{a}=2 \bar{j}-\bar{k}$$ and $$\overline{\mathbf{b}}=-\overline{\mathbf{i}}+\overline{\mathbf{k}}$$ as adjacent sides.
Solution:
Vector area of the parallelogram having $$\bar{a}=2 \bar{j}-\bar{k}$$ and $$\overline{\mathbf{b}}=-\overline{\mathbf{i}}+\overline{\mathbf{k}}$$ as adjacent sides.

Question 11.
Find the area of the parallelogram, whose diagonals are $$3 \overline{\mathbf{i}}+\overline{\mathbf{j}}-2 \overline{\mathbf{k}}$$ and $$\bar{i}-3 \bar{j}+4 \bar{k}$$.
Solution:

Question 12.
Find the area of the triangle having $$3 \bar{i}+4 \bar{j}$$ and $$-5 \bar{i}+7 \bar{j}$$ as two of its sides.
Solution:

Question 13.
Find unit vector perpendicular to the plane determined by the vectors $$\bar{a}=4 \bar{i}+3 \bar{j}-\bar{k}$$ and $$\overline{\mathbf{b}}=2 \tilde{i}-6 \overline{\mathbf{j}}-3 \overline{\mathbf{k}}$$.
Solution:

Question 14.
Find the area of the triangle whose vertices are A(1, 2, 3), B(2, 3, 1) and C(3, 1, 2).
Solution:
Suppose $$\bar{i}, \bar{j}, \bar{k}$$ are unit vectors along the co-ordinate axes.
Position vectors of A, B, C are

II.

Question 1.
If $$\overline{\mathbf{a}}+\overline{\mathbf{b}}+\overline{\mathbf{c}}=\overline{\mathbf{0}}$$, then prove that $$\overline{\mathbf{a}} \times \overline{\mathbf{b}}=\overline{\mathbf{b}} \times \overline{\mathbf{c}}=\overline{\mathbf{c}} \times \overline{\mathbf{a}}$$.
Solution:

Question 2.
If $$\overline{\mathbf{a}}=2 \bar{i}+\bar{j}-\bar{k}, \quad \bar{b}=-\bar{i}+2 \bar{j}-4 \bar{k}$$ and $$\overline{\mathbf{c}}=\overline{\mathbf{i}}+\mathbf{j}+\overline{\mathbf{k}}$$, then find $$(\bar{a} \times \bar{b}) \cdot(\bar{b} \times \bar{c})$$.
Solution:

Question 3.
Find the vector area and the area of the parallelogram having $$\overline{\mathbf{a}}=\overline{\mathbf{i}}+2 \overline{\mathbf{j}}-\overline{\mathbf{k}}$$ and $$\overline{\mathbf{b}}=2 \overline{\mathbf{i}}-\overline{\mathbf{j}}+\mathbf{2} \overline{\mathbf{k}}$$ as adjacent sides.
Solution:

Question 4.
If $$\overline{\mathbf{a}} \times \overline{\mathbf{b}}=\overline{\mathbf{b}} \times \overline{\mathbf{c}} \neq \overline{\mathbf{0}}$$, show that, $$\overline{\mathbf{a}}+\overline{\mathbf{c}}=\mathbf{p} \overline{\mathbf{b}}$$, where p is some scalar.
Solution:

Question 5.
Let $$\overline{\mathbf{a}}$$ and $$\overline{\mathbf{b}}$$ be vectors, satisfying $$|\overline{\mathbf{a}}|=|\overline{\mathbf{b}}|=5$$ and $$(\bar{a}, \bar{b})=45^{\circ}$$. Find the area of the triangle having $$\overline{\mathbf{a}}-\mathbf{2} \overline{\mathbf{b}}$$ and $$3 \bar{a}+2 \bar{b}$$ as two of its sides.
Solution:

Question 6.
Find the vector having magnitude ?6 units and perpendicular to both $$\mathbf{2} \overline{\mathbf{i}}-\overline{\mathbf{k}}$$ and $$\mathbf{3} \overline{\mathbf{i}}-\overline{\mathbf{j}}-\overline{\mathbf{k}}$$.
Solution:

Question 7.
Find a unit vector perpendicular to the plane determined by the points P(1, -1, 2), Q(2, 0, -1) and R(0, 2, 1).
Solution:

Question 8.
If $$\overline{\mathbf{a}} \cdot \overline{\mathbf{b}}=\overline{\mathbf{a}} \cdot \overline{\mathbf{c}}$$ and $$\overline{\mathbf{a}} \times \overline{\mathrm{b}}=\overline{\mathrm{a}} \times \overline{\mathrm{c}}, \overline{\mathbf{a}} \neq 0$$, then show that $$\overline{\mathbf{b}}=\overline{\mathbf{c}}$$.
Solution:

Question 9.
Find a vector of magnitude 3 and perpendicular to both the vector $$\overline{\mathbf{b}}=2 \bar{i}-2 \bar{j}+\bar{k}$$ and $$\bar{c}=2 \bar{i}+2 \bar{j}+3 \bar{k}$$.
Solution:

Question 10.
If $$|\overline{\mathbf{a}}|$$ = 13, $$|\overline{\mathbf{b}}|$$ = 5 and $$\overline{\mathbf{a}} \cdot \overline{\mathbf{b}}=\mathbf{6 0}$$, then find $$|\overline{\mathbf{a}} \times \overline{\mathbf{b}}|$$.
Solution:

Question 11.
Find a unit vector perpendicular to the plane passing through the points (1, 2, 3), (2, -1, 1) and (1, 2, -4).
Solution:
Let ‘O’ be the origin and let A, B, C be the given points.

III.

Question 1.
If $$\overline{\mathbf{a}}$$, $$\overline{\mathbf{b}}$$ and $$\overline{\mathbf{c}}$$ represent the vertices A, B and C respectively of ∆ABC, then prove that $$|(\overline{\mathbf{a}} \times \overline{\mathbf{b}})+(\overline{\mathbf{b}} \times \overline{\mathbf{c}})+(\overline{\mathbf{c}} \times \overline{\mathbf{a}})|$$ is twice the area of ∆ABC.
Solution:

Question 2.
If $$\overline{\mathbf{a}}=2 \overline{\mathbf{i}}+3 \overline{\mathbf{j}}+4 \overline{\mathbf{k}}, \overline{\mathbf{b}}=\overline{\mathbf{i}}+\overline{\mathbf{j}}-\overline{\mathbf{k}}$$ and $$\overline{\mathbf{c}}=\overline{\mathbf{i}}-\overline{\mathbf{j}}+\overline{\mathbf{k}}$$, then compute $$\overline{\mathbf{a}} \times(\overline{\mathbf{b}} \times \overline{\mathbf{c}})$$ and verify that it is perpendicular to $$\bar{a}$$.
Solution:

Question 3.
If $$\overline{\mathbf{a}}=7 \overline{\mathbf{i}}-2 \overline{\mathbf{j}}+3 \overline{\mathbf{k}}, \overline{\mathbf{b}}=2 \overline{\mathbf{i}}+8 \overline{\mathbf{k}}$$ and $$\overline{\mathbf{c}}=\overline{\mathbf{i}}+\overline{\mathbf{j}}+\overline{\mathbf{k}}$$ then compute $$\overline{\mathbf{a}} \times \mathbf{b}, \overline{\mathbf{a}} \times \overline{\mathbf{c}}$$ and $$\bar{a} \times(\bar{b}+\bar{c})$$. Verify whether cross product is distributive over vector addition.
Solution:

Question 4.
If $$\overline{\mathbf{a}}=\overline{\mathbf{i}}+\overline{\mathbf{j}}+\overline{\mathbf{k}}, \overline{\mathbf{c}}=\overline{\mathbf{j}}-\overline{\mathbf{k}}$$, then find vector $$\overline{\mathbf{b}}$$ such that $$\overline{\mathbf{a}} \times \overline{\mathbf{b}}=\overline{\mathbf{c}}$$ and $$\bar{a} \cdot \bar{b}=3$$.
Solution:

Question 5.
$$\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}$$ are three vectors of equal magnitudes and each of them is inclined at an angle of 60° to the others. If $$|\bar{a}+\bar{b}+\bar{c}|=\sqrt{6}$$, then find $$|\overline{\mathbf{a}}|$$.
Solution:

Question 6.
For any two vectors $$\bar{a}$$ and $$\bar{b}$$, show that $$\left(1+|\bar{a}|^{2}\right)\left(1+|\bar{b}|^{2}\right)$$ = $$|\mathbf{1}-\overline{\mathbf{a}} \cdot \overline{\mathbf{b}}|^{2}+|\overline{\mathbf{a}}+\overline{\mathbf{b}}+\overline{\mathbf{a}} \times \overline{\mathbf{b}}|^{2}$$
Solution:

Question 7.
If $$\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}$$ are unit vectors such that $$\overline{\mathbf{a}}$$ is perpendicular to the plane of $$\overline{\mathbf{b}}, \overline{\mathbf{c}}$$ and the angle between $$\overline{\mathbf{b}}$$ and $$\overline{\mathbf{c}}$$ is $$\frac{\pi}{3}$$, then find $$|\bar{a}+\bar{b}+\bar{c}|$$.
Solution:
Given that $$|\bar{a}|=|\bar{b}|=|\bar{c}|=1$$

Question 8.
$$\overline{\mathbf{a}}=3 \overline{\mathbf{i}}-\overline{\mathbf{j}}+2 \overline{\mathbf{k}}, \overline{\mathbf{b}}=-\overline{\mathbf{i}}+3 \overline{\mathbf{j}}+2 \overline{\mathbf{k}}$$, $$\bar{c}=4 \bar{i}+5 \bar{j}-2 \bar{k}$$ and $$\bar{d}=\bar{i}+3 \bar{j}+5 \bar{k}$$ then compute the following.
(i) $$(\overline{\mathbf{a}} \times \overline{\mathbf{b}}) \times(\bar{c} \times \bar{d})$$ and
(ii) $$(\overline{\mathbf{a}} \times \overline{\mathbf{b}}) \cdot \overline{\mathbf{c}}-(\overline{\mathbf{a}} \times \overline{\mathbf{d}}) \cdot \overline{\mathbf{b}}$$
Solution: