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

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

I.

Question 1.
Find the angle between the vectors $$\bar{i}+2 \bar{j}+3 \bar{k}$$ and $$3 \bar{i}-\bar{j}+2 \bar{k}$$.
Solution:
Let $$\overline{\mathrm{a}}=\overline{\mathrm{i}}+2 \overline{\mathrm{j}}+3 \overline{\mathrm{k}}$$ and $$\overline{\mathrm{b}}=3 \overline{\mathrm{i}}-\overline{\mathrm{j}}+2 \overline{\mathrm{k}}$$ and ‘θ’ be the angle between them (i.e.,) $$(\bar{a}, \bar{b})$$ = θ Question 2.
If the vectors $$\mathbf{2} \overline{\mathbf{i}}+\lambda \overline{\mathbf{j}}-\overline{\mathbf{k}}$$ and $$4 \bar{i}-2 \bar{j}+2 \bar{k}$$ are perpendicular to each other, then find λ.
Solution:  Question 3.
For what values of λ, the vectors $$\overline{\mathbf{i}}-\lambda \overline{\mathbf{j}}+2 \overline{\mathbf{k}}$$ and $$8 \overline{\mathbf{i}}+6 \overline{\mathbf{j}}-\overline{\mathbf{k}}$$ are at right angles?
Solution: Question 4.
$$\overline{\mathbf{a}}=2 \overline{\mathbf{i}}-\overline{\mathbf{j}}+\overline{\mathbf{k}}, \overline{\mathbf{b}}=\overline{\mathbf{i}}-3 \overline{\mathbf{j}}-5 \overline{\mathbf{k}}$$. Find the vector C such that $$\overline{\mathbf{a}}$$, $$\overline{\mathbf{b}}$$ and $$\overline{\mathbf{c}}$$ form the sides of a triangle.
Solution: Question 5.
Find the angle between the planes $$\bar{r} \cdot(2 \bar{i}-\bar{j}+2 \bar{k})=3$$ and $$\overline{\mathrm{r}} \cdot(3 \overline{\mathrm{i}}+6 \bar{j}+\bar{k})=4$$.
Solution: Question 6.
Let $$\overline{\mathbf{e}}_{1}$$ and $$\overline{\mathbf{e}}_{2}$$ be unit vectors makingangle θ. If $$\frac{1}{2}\left|\bar{e}_{1}-\bar{e}_{2}\right|=\sin \lambda \theta$$, then find λ.
Solution:   Question 7.
Let $$\overline{\mathbf{a}}=\overline{\mathbf{i}}+\overline{\mathbf{j}}+\overline{\mathbf{k}}$$ and $$\overline{\mathbf{b}}=\mathbf{2} \overline{\mathbf{i}}+3 \overline{\mathbf{j}}+\overline{\mathbf{k}}$$. Find
(i) The projection vector of $$\overline{\mathbf{b}}$$ on $$\overline{\mathbf{a}}$$ and its magnitude.
(ii) The vector components of $$\overline{\mathbf{b}}$$ in the direction of a and perpendicular to $$\overline{\mathbf{a}}$$.
Solution: Question 8.
Find the equation of the plane through the point (3, -2, 1) and perpendicular to the vector (4, 7, -4).
Solution: Question 9.
If $$\overline{\mathbf{a}}=2 \bar{i}+2 \bar{j}-3 \bar{k}$$; $$\overline{\mathbf{b}}=3 \overline{\mathbf{i}}-\overline{\mathbf{j}}+2 \overline{\mathbf{k}}$$, then find the angle between $$2 \overline{\mathbf{a}}+\overline{\mathbf{b}}$$ and $$\bar{a}+2 \bar{b}$$.
Solution: II.

Question 1.
Find unit vector parallel to the XOY- plane and perpendicular to the vector $$4 \bar{i}-3 \bar{j}+\bar{k}$$.
Solution:
Any vector parallel to XOY-plane will be of the form $$p \bar{i}+q \bar{j}$$
∴ The vector parallel to the XOY-plane and perpendicular to the vector $$4 \bar{i}-3 \bar{j}+\bar{k}$$ is $$3 \bar{i}+4 \bar{j}$$
Its magnitude = $$|3 \bar{i}+\overline{4 j}|=\sqrt{9+16}=5$$
∴ Unit vector parallel to the XOY-plane and perpendicular to the vector $$4 \bar{i}-3 \bar{j}+\bar{k}$$ is $$\pm \frac{(3 \overline{\mathrm{i}}+4 \overline{\mathrm{j}})}{5}$$ Question 2.
If $$\overline{\mathbf{a}}+\overline{\mathbf{b}}+\overline{\mathrm{c}}=0,|\overline{\mathbf{a}}|=3,|\overline{\mathbf{b}}|=5$$ and $$|\bar{c}|=7$$, then find the angle between $$\overline{\mathbf{a}}$$ and $$\overline{\mathbf{b}}$$.
Solution: Question 3.
If $$|\overline{\mathbf{a}}|$$ = 2, $$|\overline{\mathbf{b}}|$$ = 3 and $$|\overline{\mathbf{c}}|$$ = 4 and each of $$\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}$$ is perpendicular to the sum of the other two vectors, then find the magnitude of $$\overline{\mathbf{a}}+\overline{\mathbf{b}}+\overline{\mathbf{c}}$$.
Solution:  Question 4.
Find the equation of the plane passing through the point $$\overline{\mathbf{a}}=\mathbf{2} \overline{\mathbf{i}}+3 \bar{j}-\overline{\mathbf{k}}$$ and perpendicular to the vector $$3 \bar{i}-2 \bar{j}-2 \bar{k}$$ and the distance of this plane from the origin.
Solution:
Equation of the plane passing through the point $$\overline{\mathbf{a}}=\mathbf{2} \overline{\mathbf{i}}+3 \bar{j}-\overline{\mathbf{k}}$$ and perpendicular to the vector $$\bar{n}=3 \bar{i}-2 \bar{j}-2 \bar{k}$$ is  Question 5.
$$\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}$$ and $$\overline{\mathbf{d}}$$ are the position vectors of four coplanar points such that $$(\mathbf{a}-\overline{\mathbf{d}}) \cdot(\bar{b}-\bar{c})=(\bar{b}-\bar{d}) \cdot(\bar{c}-\bar{a})=0$$. Show that the point $$\bar{d}$$ represents the orthocentre of the triangle with $$\bar{a}$$, $$\bar{b}$$ and $$\bar{c}$$ as its vertices.
Solution:
Position vectors of A, B, C, D are $$\bar{a}$$, $$\bar{b}$$, $$\bar{c}$$ and $$\bar{d}$$ respectively.  ⇒ BD is perpendicular to AC
∴ BD is another altitude of ∆ABC.
Altitudes AD and BD intersect at D.
∴ D is the orthocentre of ∆ABC.

III.

Question 1.
Show that the points (5, -1, 1), (7, -4, 7), (1, -6, 10) and (-1, -3, 4) are the vertices of a rhombus.
Solution:
Let A(5, -1, 1), B(7, -4, 7), C(1, -6, 10) and D(-1, -3, 4) are the given points. ∵ AB = BC = CD = DA = 7 units
AC ≠ BD
∴ A, B, C, D points are the vertices of a rhombus. Question 2.
Let $$\bar{a}=4 \bar{i}+5 \bar{j}-\bar{k}, \quad \bar{b}=\bar{i}-4 \bar{j}+5 \bar{k}$$ and $$\overline{\mathbf{c}}=\mathbf{3} \overline{\mathbf{i}}+\overline{\mathbf{j}}-\overline{\mathbf{k}}$$. Find the vector which is perpendicular to both $$\overline{\mathbf{a}}$$ and $$\overline{\mathbf{b}}$$ and whose magnitude is twenty one times the magnitude of $$\overline{\mathbf{c}}$$.
Solution:  Question 3.
G is the centroid of ΔABC and a, b, c are the lengths of the sides BC, CA and AB respectively prove that a2 + b2 + c2 = 3 (OA2 + OB2 + OC2) – 9(OG)2 where O is any point.
Solution:
Given that $$\overline{\mathrm{BC}}=\overline{\mathrm{a}}, \overline{\mathrm{CA}}=\overline{\mathrm{b}}, \overline{\mathrm{AB}}=\overline{\mathrm{c}}$$
Let ‘O’ be the origin and let $$\overline{\mathrm{OA}}=\overline{\mathrm{p}}, \overline{\mathrm{OB}}=\overline{\mathrm{q}} \text { and } \overline{\mathrm{OC}}=\overline{\mathrm{r}}$$
Then P.V. of the centroid of ΔABC is From (1) and (2)
∴ a2 + b2 + c2 = 3(OA2 + OB2 + OC2) – 9(OG)2. Question 4.
A line makes angles θ1, θ2, θ3, and θ4 with the diagonals of a cube. Show that cos2θ1 + cos2θ2 + cos2θ3 + cos2θ4 = $$\frac{4}{3}$$.
Solution:   