SATHEE: Vectors 5 Question 7

Vectors 5 Question 7

7. Let $\vec{b} = 4 \hat{i} + 3 \hat{j}$ and $\vec{c}$ be two vectors perpendicular to each other in the $X Y$ -plane. All vectors in the same plane having projections 1 and 2 along $\vec{b}$ and $\vec{c}$ , respectively are given by…. .

(1987, 2M)

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Answer:

Correct Answer: 7. $2 \hat{i} - \hat{j}$

Solution:

Let $\vec{c} = a \hat{i} + b \hat{j}$

Since, $\vec{b}$ and $\vec{c}$ are perpendiculars to each other. Then,

$\begin{aligned} \vec{b} \cdot \vec{c} = 0 \Rightarrow (4 \hat{i} + 3 \hat{j}) \cdot (a \hat{i} + b \hat{j}) = 0 \\ \Rightarrow 4 a + 3 b = 0 \Rightarrow a : b = 3 : - 4 \\ ∴ \vec{c} = λ (3 \hat{i} - 4 \hat{j}), where λ is constant of ratio. \end{aligned}$

Let the required vectors be $\vec{a} = p \hat{i} + q \hat{j}$

Projection of $\vec{a}$ on $\vec{b}$ is $\frac{\vec{a} \cdot \vec{b}}{| \vec{b} |}$

$∴ 1 = \frac{4 p + 3 q}{5} \Rightarrow 4 p + 3 q = 5$

Also, projection of $\vec{a}$ on $\vec{c}$ is $\frac{\vec{a} \cdot \vec{c}}{| \vec{c} |}$

$\Rightarrow 2 = \frac{3 λ p - 4 λ q}{5 λ} \Rightarrow 3 p - 4 q = 10$

On solving above equations, we get $p = 2, q = - 1$

$∴ \vec{c} = 2 \hat{i} - \hat{j}$

Vectors 5 Question 7

7. Let $\vec{b} = 4 \hat{i} + 3 \hat{j}$ and $\vec{c}$ be two vectors perpendicular to each other in the $X Y$ -plane. All vectors in the same plane having projections 1 and 2 along $\vec{b}$ and $\vec{c}$ , respectively are given by…. .

Answer:

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Vectors 5 Question 7

7. Let b→=4i^+3j^ and c→ be two vectors perpendicular to each other in the XY-plane. All vectors in the same plane having projections 1 and 2 along b→ and c→, respectively are given by…. .

Answer:

Engineering Preparation

Medical Preparation

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Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

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NCERT Exercise Video Solution

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7. Let $\vec{b} = 4 \hat{i} + 3 \hat{j}$ and $\vec{c}$ be two vectors perpendicular to each other in the $X Y$ -plane. All vectors in the same plane having projections 1 and 2 along $\vec{b}$ and $\vec{c}$ , respectively are given by…. .