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JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Vector Algebra

Let

α = 4 \hat{i} + 3 \hat{j} + 5 \hat{k}

and

β = \hat{i} + 2 \hat{j} - 4 \hat{k}

. Let

β_{1}

be parallel to

α

and

β_{2}

be perpendicular to

α

. If

β = β_{1} + β_{2}

, then the value of

5 β_{2} \cdot (\hat{i} + \hat{j} + \hat{k})

is

(A)6
(B)11
(C)7✓ Correct
(D)9

Answer: (C)

Solution

Resolving

β

along and across

α

, the parallel part is

β_{1} = \frac{β \cdot α}{∣ α ∣ ^{2}} α

. Here

β \cdot α = 4 + 6 - 20 = - 10

and

∣ α ∣^{2} = 16 + 9 + 25 = 50

, so

β_{1} = - \frac{1}{5} α

. The perpendicular part is

β_{2} = β - β_{1} = (\hat{i} + 2 \hat{j} - 4 \hat{k}) + \frac{1}{5} (4 \hat{i} + 3 \hat{j} + 5 \hat{k}) = \frac{9}{5} \hat{i} + \frac{13}{5} \hat{j} - 3 \hat{k}

. Then

5 β_{2} = 9 \hat{i} + 13 \hat{j} - 15 \hat{k}

, and

5 β_{2} \cdot (\hat{i} + \hat{j} + \hat{k}) = 9 + 13 - 15 = 7

.

JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Single correct option

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