SATHEE: Vectors 1 Question 6

Vectors 1 Question 6

7. Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{c} = \hat{i} - \hat{j} - \hat{k}$ be three vectors. A vector $\vec{v}$ in the plane of $\vec{a}$ and $\vec{b}$ , whose projection on $\vec{c}$ is $\frac{1}{\sqrt{3}}$ , is given by

(2011)

(a) $\hat{i} - 3 \hat{j} + 3 \hat{k}$

(b) $- 3 \hat{i} - 3 \hat{j} - \hat{k}$

(d) $\hat{i} + 3 \hat{j} - 3 \hat{k}$

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Answer:

Correct Answer: 7. (c)

Solution:

Let $\vec{v} = \vec{a} + λ \vec{b}$

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

Projection of $\vec{v}$ on $\vec{c} = \frac{1}{\sqrt{3}} \Rightarrow \frac{\vec{v} \cdot \vec{c}}{| \vec{c} |} = \frac{1}{\sqrt{3}}$

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

$\Rightarrow 1 + λ - 1 + λ - 1 - λ = 1$

$\Rightarrow λ - 1 = 1 \Rightarrow λ = 2$

$∴ \vec{v} = 3 \hat{i} - \hat{j} + 3 \hat{k}$

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Vectors 1 Question 6

7. Let a→=i^+j^+k^,b→=i^−j^+k^ and c→=i^−j^−k^ be three vectors. A vector v→ in the plane of a→ and b→, whose projection on c→ is 13, is given by

Answer:

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7. Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{c} = \hat{i} - \hat{j} - \hat{k}$ be three vectors. A vector $\vec{v}$ in the plane of $\vec{a}$ and $\vec{b}$ , whose projection on $\vec{c}$ is $\frac{1}{\sqrt{3}}$ , is given by