Representation and Components of Vector | Vectors previous year questions

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Topic Question Set

Q 1 :
A vector has magnitude same as that of $\vec{A} = 3 \hat{i} + 4 \hat{j}$ and is parallel to $\vec{B} = 4 \hat{i} + 3 \hat{j}$ . The $x$ and $y$ components of this vector in first quadrant are $x$ and 3 respectively where $x =$ _________ . [2024]

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(4) Given $\vec{A} = 3 i + 4 j,$ $\vec{B} = 4 i + 3 j,$

$| A | = \sqrt{3^{2} + 4^{2}} = 5,$ $\hat{B} = \frac{4 i + 3 j}{5}, \frac{\vec{B}}{| B |} = \hat{B}$

Now, as per question, $\vec{C} = | A | \hat{B},$ mag of $\vec{A}$ and direction of B

$= (5) \frac{(4 i + 3 j)}{5} = 4 i + 3 j, 4 i + 3 j,$ $∴ x = 4$

Q 2 :
The position vector of a moving body at any instant of time is given as $\vec{r} = (5 t^{2} \hat{i} - 5 t \hat{j}) m$ . The magnitude and direction of velocity at t = 2 s is, [2025]

$5 \sqrt{15}$ m/s, making an angle of $\tan^{- 1}$ 4 with –ve Y axis
$5 \sqrt{15}$ m/s, making an angle of $\tan^{- 1}$ 4 with +ve X axis
$5 \sqrt{17}$ m/s, making an angle of $\tan^{- 1}$ 4 with –ve Y axis
$5 \sqrt{17}$ m/s, making an angle of $\tan^{- 1}$ 4 with +ve X axis

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(3)

$\vec{r} = 5 t^{2} \hat{i} - 5 \hat{j}$

$\vec{v} = 10 t \hat{i} - 5 \hat{j}$

$\vec{v} = 20 \hat{i} - 5 \hat{j} at t = 2 \sec$

$| \vec{v} | = 5 \sqrt{17} m / s$

$\tan θ = \frac{20}{5} = 4$

$θ = \tan^{- 1}$ 4 with –ve Y-axis

Q 3 :
A light wave is propagating with plane wave fronts of the type x + y + z = constant. The angle made by the direction of wave propagation with the x-axis is: [2025]

$\cos^{- 1} (\frac{1}{\sqrt{3}})$
$\cos^{- 1} (\frac{2}{3})$
$\cos^{- 1} (\frac{1}{3})$
$\cos^{- 1} (\sqrt{\frac{2}{3}})$

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(1)

Direction of propagation $\vec{A} = \hat{i} + \hat{j} + \hat{k}$

The direction of propagation of light is perpendicular to the wave front and is symmetric about x, y and z-axis.

Direction cosine associated with x-axis,

$\cos α = (\frac{A_{x}}{\sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}}})$

$\cos α = (\frac{1}{\sqrt{1 + 1 + 1}}) \Rightarrow α = \cos^{- 1} (\frac{1}{\sqrt{3}})$

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