SATHEE: Vectors 5 Question 2

Vectors 5 Question 2

2. Let $a = 2 \hat{i} + λ_{1} \hat{j} + 3 \hat{k}, b = 4 \hat{i} + (3 - λ_{2}) \hat{j} + 6 \hat{k}$ and $c = 3 \hat{i} + 6 \hat{j} + (λ_{3} - 1) \hat{k}$ be three vectors such that $b = 2 a$ and $a$ is perpendicular to $c$ . Then a possible value of $(λ_{1}, λ_{2}, λ_{3})$ is

(2019 Main, 10 Jan I)

(a) $(1, 3, 1)$

(b) $(1, 5, 1)$

(d) $(\frac{1}{2}, 4, - 2)$

Show Answer

Answer:

Correct Answer: 2. (c)

Solution:

We have, $a = 2 \hat{i} + λ_{1} \hat{j} + 3 \hat{k}; b = 4 \hat{i} + (3 - λ_{2}) \hat{j} + 6 \hat{k}$

and $c = 3 \hat{i} + 6 \hat{j} + (λ_{3} - 1) \hat{k}$ ,

such that $b = 2 a$

Now, $b = 2 a$

$\Rightarrow 4 \hat{i} + (3 - λ_{2}) \hat{j} + 6 \hat{k} = 2 (2 \hat{i} + λ_{1} \hat{j} + 3 \hat{k})$

$\Rightarrow 4 \hat{i} + (3 - λ_{2}) \hat{j} + 6 \hat{k} = 4 \hat{i} + 2 λ_{1} \hat{j} + 6 \hat{k}$

$\Rightarrow (3 - 2 λ_{1} - λ_{2}) \hat{j} = 0$

$\Rightarrow 3 - 2 λ_{1} - λ_{2} = 0$

$\Rightarrow 2 λ_{1} + λ_{2} = 3$

Also, as a is perpendicular to $c$ , therefore $a \cdot c = 0$

$\Rightarrow (2 \hat{i} + λ_{1} \hat{j} + 3 \hat{k}) \cdot (3 \hat{i} + 6 \hat{j} + (λ_{3} - 1) \hat{k}) = 0$

$\Rightarrow 6 + 6 λ_{1} + 3 (λ_{3} - 1) = 0$

$\Rightarrow 6 λ_{1} + 3 λ_{3} + 3 = 0$

$\Rightarrow 2 λ_{1} + λ_{3} = - 1$

Now, from Eq. (i), $λ_{2} = 3 - 2 λ_{1}$ and from Eq. (ii)

$λ_{3} = - 2 λ_{1} - 1$

$∴ (λ_{1}, λ_{2}, λ_{3}) \equiv (λ_{1}, 3 - 2 λ_{1}, - 2 λ_{1} - 1)$

If $λ_{1} = - \frac{1}{2}$ , then

$λ_{2} = 4, and λ_{3} = 0$

Thus, a possible value of $(λ_{1}, λ_{2}, λ_{3}) = (- \frac{1}{2}, 4, 0)$

Vectors 5 Question 2

2. Let $a = 2 \hat{i} + λ_{1} \hat{j} + 3 \hat{k}, b = 4 \hat{i} + (3 - λ_{2}) \hat{j} + 6 \hat{k}$ and $c = 3 \hat{i} + 6 \hat{j} + (λ_{3} - 1) \hat{k}$ be three vectors such that $b = 2 a$ and $a$ is perpendicular to $c$ . Then a possible value of $(λ_{1}, λ_{2}, λ_{3})$ is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Vectors 5 Question 2

2. Let a=2i^+λ1j^+3k^,b=4i^+(3−λ2)j^+6k^ and c=3i^+6j^+(λ3−1)k^ be three vectors such that b=2a and a is perpendicular to c. Then a possible value of (λ1,λ2,λ3) 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

2. Let $a = 2 \hat{i} + λ_{1} \hat{j} + 3 \hat{k}, b = 4 \hat{i} + (3 - λ_{2}) \hat{j} + 6 \hat{k}$ and $c = 3 \hat{i} + 6 \hat{j} + (λ_{3} - 1) \hat{k}$ be three vectors such that $b = 2 a$ and $a$ is perpendicular to $c$ . Then a possible value of $(λ_{1}, λ_{2}, λ_{3})$ is