JEE Mathematics 2023 Question with Solution

Given: $\vec{a} = 4\hat{i} + 3\hat{j}$, $\vec{b} = 3\hat{i} - 4\hat{j} + 5\hat{k}$. Let $\vec{c} = x\hat{i} + y\hat{j} + z\hat{k}$.

Find: The projection of $\vec{c}$ on $\vec{b}$.

First compute $\vec{a} \times \vec{b}$:

Using $(\vec{a} \times \vec{b}) \cdot \vec{c} + 25 = 0$,

Using $\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = 4$,

The projection of $\vec{c}$ on $\vec{a}$ is $1$, so

Solving equations $(1)$, $(2)$, and $(3)$, we get

Now compute the projection of $\vec{c}$ on $\vec{b}$:

Therefore, the correct option is A.

Given: The three conditions give three linear equations in $x, y, z$ after taking $\vec{c} = x\hat{i} + y\hat{j} + z\hat{k}$.

Find: The scalar projection of $\vec{c}$ on $\vec{b}$.

The shortcut is to immediately convert each vector condition into a scalar equation:

• From the scalar triple product condition: $3x - 4y - 5z = -5$
• From $\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = 4$: $x + y + z = 4$
• From projection on $\vec{a}$ being $1$: $4x + 3y = 5$

Solving these gives

so

Then evaluate only the required quantity:

Thus the projection of $\vec{c}$ on $\vec{b}$ is $\frac{5}{\sqrt{2}}$.