CUET UG 2025 — Mathematics Algebra

The projection of vector $\vec{a}$a on vector $\vec{b}$b is given by:

Projection $= \dfrac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$=∣b∣a⋅b​

where $\vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}$a=2i^−j^​+3k^ and $\vec{b} = 3\hat{i} + 2\hat{j} + 6\hat{k}$b=3i^+2j^​+6k^.

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The dot product $\vec{a} \cdot \vec{b}$a⋅b:

$$\vec{a} \cdot \vec{b} = (2)(3) + (-1)(2) + (3)(6)$$a⋅b=(2)(3)+(−1)(2)+(3)(6)

$$= 6 - 2 + 18$$=6−2+18

$$= 22$$=22

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The magnitude of $\vec{b}$b:

$$|\vec{b}| = \sqrt{(3)^2 + (2)^2 + (6)^2}$$∣b∣=(3)2+(2)2+(6)2​

$$= \sqrt{9 + 4 + 36}$$=9+4+36​

$$= \sqrt{49}$$=49​

$$= 7$$=7

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The projection:

Projection $= \dfrac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$=∣b∣a⋅b​

$$= \dfrac{22}{7}$$=722​

Therefore, the projection of the vector $2\hat{i} - \hat{j} + 3\hat{k}$2i^−j^​+3k^ on the vector $3\hat{i} + 2\hat{j} + 6\hat{k}$3i^+2j^​+6k^ is $\dfrac{22}{7}$722​.