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

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
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) I Q1

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:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) I Q2

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

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:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) I Q3

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:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) I Q4

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:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) I Q5

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:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) I Q6
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) I Q6.1

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

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:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) I Q7

Question 8.
Find the equation of the plane through the point (3, -2, 1) and perpendicular to the vector (4, 7, -4).
Solution:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) I Q8

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:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) I Q9

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 \(\pm \frac{(3 \overline{\mathrm{i}}+4 \overline{\mathrm{j}})}{5}\)

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

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:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) II Q2

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:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) II Q3
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) II Q3.1

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
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) II Q4

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

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.
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) II Q5
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) II Q5.1
⇒ 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.
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) III Q1
∵ AB = BC = CD = DA = 7 units
AC ≠ BD
∴ A, B, C, D points are the vertices of a rhombus.

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

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:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) III Q2
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) III Q2.1

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
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) III Q3
From (1) and (2)
∴ a2 + b2 + c2 = 3(OA2 + OB2 + OC2) – 9(OG)2.

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

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:
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) III Q4
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) III Q4.1
Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a) III Q4.2

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