JEE Mathematics 2023 Question with Solution

Given: $\vec{\alpha} = 4\hat{i} + 3\hat{j} + 5\hat{k}$ and $\vec{\beta} = \hat{i} + 2\hat{j} - 4\hat{k}$.

Find: $5\vec{\beta}_2 \cdot (\hat{i} + \hat{j} + \hat{k})$ where $\vec{\beta}_1$ is parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ is perpendicular to $\vec{\alpha}$.

The component of $\vec{\beta}$ parallel to $\vec{\alpha}$ is

$$\vec{\beta}_1 = \frac{\vec{\alpha} \cdot \vec{\beta}}{\vec{\alpha} \cdot \vec{\alpha}}\vec{\alpha}$$

Now,

$$\vec{\alpha} \cdot \vec{\beta} = (4)(1) + (3)(2) + (5)(-4) = 4 + 6 - 20 = -10$$

and

$$\vec{\alpha} \cdot \vec{\alpha} = 4^2 + 3^2 + 5^2 = 16 + 9 + 25 = 50$$

Therefore,

$$\vec{\beta}_1 = \frac{-10}{50}\vec{\alpha} = -\frac{1}{5}(4\hat{i} + 3\hat{j} + 5\hat{k}) = -\frac{4}{5}\hat{i} - \frac{3}{5}\hat{j} - \hat{k}$$

Now,

$$\vec{\beta}_2 = \vec{\beta} - \vec{\beta}_1$$

So,

$$\vec{\beta}_2 = (\hat{i} + 2\hat{j} - 4\hat{k}) - \left(-\frac{4}{5}\hat{i} - \frac{3}{5}\hat{j} - \hat{k}\right)$$ $$\vec{\beta}_2 = \hat{i} + \frac{4}{5}\hat{i} + 2\hat{j} + \frac{3}{5}\hat{j} - 4\hat{k} + \hat{k}$$ $$\vec{\beta}_2 = \frac{9}{5}\hat{i} + \frac{13}{5}\hat{j} - 3\hat{k}$$

Hence,

$$5\vec{\beta}_2 = 9\hat{i} + 13\hat{j} - 15\hat{k}$$

Now take dot product with $\hat{i} + \hat{j} + \hat{k}$:

$$5\vec{\beta}_2 \cdot (\hat{i} + \hat{j} + \hat{k}) = 9(1) + 13(1) - 15(1)$$ $$= 9 + 13 - 15 = 7$$

Therefore, the correct option is C.

Given: $\vec{\beta} = \vec{\beta}_1 + \vec{\beta}_2$ with $\vec{\beta}_1 \parallel \vec{\alpha}$ and $\vec{\beta}_2 \perp \vec{\alpha}$.

Find: the required scalar value using vector decomposition.

Use projection of $\vec{\beta}$ on $\vec{\alpha}$ to obtain the parallel part. Then subtract it from $\vec{\beta}$ to get the perpendicular part.

The projection factor is

$$\frac{\vec{\alpha} \cdot \vec{\beta}}{|\vec{\alpha}|^2} = \frac{-10}{50} = -\frac{1}{5}$$

So the parallel component is

$$\vec{\beta}_1 = -\frac{1}{5}\vec{\alpha}$$

This gives

$$\vec{\beta}_1 = -\frac{4}{5}\hat{i} - \frac{3}{5}\hat{j} - \hat{k}$$

Subtract from $\vec{\beta}$:

$$\vec{\beta}_2 = \left(1 + \frac{4}{5}\right)\hat{i} + \left(2 + \frac{3}{5}\right)\hat{j} + (-4 + 1)\hat{k}$$ $$\vec{\beta}_2 = \frac{9}{5}\hat{i} + \frac{13}{5}\hat{j} - 3\hat{k}$$

Multiply by $5$:

$$5\vec{\beta}_2 = 9\hat{i} + 13\hat{j} - 15\hat{k}$$

Now dot with $\hat{i} + \hat{j} + \hat{k}$ by adding corresponding components:

$$9 + 13 - 15 = 7$$

Therefore, the required value is $7$, so the correct option is C.