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JEE Main · 2023 · 13 Apr 2023 · Shift 1 · Vector Algebra

Let

a = 3 \hat{i} + \hat{j} - \hat{k}

and

c = 2 \hat{i} - 3 \hat{j} + 3 \hat{k}

. If

b

is a vector such that

a = b \times c

and

∣ b ∣^{2} = 50

, then

72 - ∣ b + c ∣^{2}

is equal to _____.

Answer: 66

Solution

Here

∣ a ∣^{2} = 11

,

∣ c ∣^{2} = 22

and

∣ b ∣^{2} = 50

. Since

a = b \times c

, the Lagrange identity gives

∣ a ∣^{2} = ∣ b ∣^{2} ∣ c ∣^{2} - (b \cdot c)^{2}

, so

11 = 50 \cdot 22 - (b \cdot c)^{2}

and

(b \cdot c)^{2} = 1089

, i.e.

b \cdot c = \pm 33

. Now

∣ b + c ∣^{2} = ∣ b ∣^{2} + ∣ c ∣^{2} + 2 b \cdot c = 72 + 2 b \cdot c

, so

72 - ∣ b + c ∣^{2} = - 2 b \cdot c = \mp 66

. Taking the modulus,

72 - ∣ b + c ∣^{2} = 66

.

JEE Main · 2023 · 13 Apr 2023 · Shift 1 · Integer answer

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