SATHEE: Vectors 2 Question 12

Vectors 2 Question 12

12. Let $\vec{A}$ be vector parallel to line of intersection of planes $P_{1}$ and $P_{2}$ through origin. $P_{1}$ is parallel to the vectors $2 \hat{j} + 3 \hat{k}$ and $4 \hat{j} - 3 \hat{k}$ and $P_{2}$ is parallel to $\hat{j} - \hat{k}$ and $3 \hat{i} + 3 \hat{j}$ , then the angle between vector $\vec{A}$ and $2 \hat{i} + \hat{j} - 2 \hat{k}$ is

(2006, 5M)

(a) $\frac{π}{2}$

(b) $\frac{π}{4}$

(d) $\frac{3 π}{4}$

Show Answer

Answer:

Correct Answer: 12. (b, d)

Solution:

Let vector $\vec{A O}$ be parallel to line of intersection of planes $P_{1}$ and $P_{2}$ through origin.

Normal to plane $p_{1}$ is

$\vec{n_{1}} = [(2 \hat{j} + 3 \hat{k}) \times (4 \hat{j} - 3 \hat{k})] = - 18 \hat{i}$

Normal to plane $p_{2}$ is

${\vec{n}}_{2} = (\hat{j} - \hat{k}) \times (3 \hat{i} + 3 \hat{j}) = 3 \hat{i} - 3 \hat{j} - 3 \hat{k}$

So, $\vec{O A}$ is parallel to $\pm ({\vec{n}}_{1} \times {\vec{n}}_{2}) = 54 \hat{j} - 54 \hat{k}$ .

$∴$ Angle between $54 (\hat{j} - \hat{k})$ and $(2 \hat{i} + \hat{j} - 2 \hat{k})$ is

$\begin{aligned} \cos θ & = \pm (\frac{54 + 108}{3 \cdot 54 \cdot \sqrt{2}}) = \pm \frac{1}{\sqrt{2}} \\ ∴ θ & = \frac{π}{4}, \frac{3 π}{4} \end{aligned}$

Hence, (b) and (d) are correct answers.

Vectors 2 Question 12

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Vectors 2 Question 12

12. Let A→ be vector parallel to line of intersection of planes P1 and P2 through origin. P1 is parallel to the vectors 2j^+3k^ and 4j^−3k^ and P2 is parallel to j^−k^ and 3i^+3j^, then the angle between vector A→ and 2i^+j^−2k^ is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu