Find the value of such that the vectors a = 2 i + j + k and b = i + 2 j + 3 k are orthogonal:

Question

Find the value of $λ$ such that the vectors $\vec{a} = 2 \hat{i} + λ \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2 \hat{j} + 3 \hat{k}$ are orthogonal:

Smart Achievers · Accepted Answer

(4)

Ans. $\frac{- 5}{2}$

Explanation:

Since, two non-zero vectors $\vec{a}$ and $\vec{b}$ are orthogonal i.e., $\vec{a} \cdot \vec{b} = 0$

$∴$ $(2 \hat{i} + λ \hat{j} + \hat{k}) \cdot (\hat{i} + 2 \hat{j} + 3 \hat{k}) = 0$

$\Rightarrow$ $2 + 2 λ + 3 = 0$

$∴$ $λ = \frac{- 5}{2}$

Solution

Q.
Find the value of $λ$ such that the vectors $\vec{a} = 2 \hat{i} + λ \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2 \hat{j} + 3 \hat{k}$ are orthogonal:

1 $0$

2 $1$

3 $\frac{3}{2}$

4 $\frac{- 5}{2}$

Ans.
(4)

Ans. $\frac{- 5}{2}$

Explanation:

Since, two non-zero vectors $\vec{a}$ and $\vec{b}$ are orthogonal i.e., $\vec{a} \cdot \vec{b} = 0$

$∴$ $(2 \hat{i} + λ \hat{j} + \hat{k}) \cdot (\hat{i} + 2 \hat{j} + 3 \hat{k}) = 0$

$\Rightarrow$ $2 + 2 λ + 3 = 0$

$∴$ $λ = \frac{- 5}{2}$

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Solution

Q. Find the value of λ such that the vectors a→=2i^+λj^+k^ and b→=i^+2j^+3k^ are orthogonal:

1 0

2 1

3 32

4 -52

Ans. (4) Ans. -52 Explanation: Since, two non-zero vectors a→ and b→ are orthogonal i.e., a→·b→=0 ∴ (2i^+λj^+k^)·(i^+2j^+3k^)=0 ⇒ 2+2λ+3=0 ∴ λ=-52