Let and Let be the vector in the plane of the vectors and , such that the length of its projection on the vector is Then is equal to [2026]

Question

Let $\vec{a} = 2 \hat{i} - \hat{j} - \hat{k}, \vec{b} = \hat{i} + 3 \hat{j} - \hat{k}$ and $\vec{c} = 2 \hat{i} + \hat{j} + 3 \hat{k} .$ Let $\vec{v}$ be the vector in the plane of the vectors $\vec{a}$ and $\vec{b}$ , such that the length of its projection on the vector $\vec{c}$ is $\frac{1}{\sqrt{14}} .$ Then $| \vec{v} |$ is equal to [2026]

Smart Achievers · Accepted Answer

(2)

$Let \vec{v} = \vec{a} + λ \vec{b}$

$\vec{v} = (2 + λ) \hat{i} + (3 λ - 1) \hat{j} - (1 + λ) \hat{k}$

$\frac{\vec{v} \cdot \vec{c}}{| \vec{c} |} = \frac{1}{\sqrt{14}}$

$\frac{2 (2 + λ) + 3 λ - 1 - 3 - 3 λ}{\sqrt{14}} = \frac{1}{\sqrt{14}}$

$4 + 2 λ - 4 = 1$

$2 λ = 1 \Rightarrow λ = \frac{1}{2}$

$\vec{v} = \frac{5}{2} \hat{i} + \frac{1}{2} \hat{j} - \frac{3}{2} \hat{k}$

$| \vec{v} |$ $= \sqrt{\frac{25 + 1 + 9}{4}} = \frac{\sqrt{35}}{2}$

Solution

1 $\frac{\sqrt{21}}{2}$

2 $\frac{\sqrt{35}}{2}$

3 $13$

4 7

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Solution

Q. Let a→=2i^-j^-k^, b→=i^+3j^-k^ and c→=2i^+j^+3k^. Let v→ be the vector in the plane of the vectors a→ and b→, such that the length of its projection on the vector c→ is 114. Then |v→| is equal to [2026]

1 212

2 352

3 13

4 7

Ans. (2) Let v→=a→+λb→ v→=(2+λ)i^+(3λ-1)j^-(1+λ)k^ v→·c→|c→|=114 2(2+λ)+3λ-1-3-3λ14=114 4+2λ-4=1 2λ=1⇒λ=12 v→=52i^+12j^-32k^ |v→|=25+1+94=352

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1 $\frac{\sqrt{21}}{2}$

2 $\frac{\sqrt{35}}{2}$

3 $13$