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Vector Algebra question

2023 · 31 Jan · Shift 1 · Q45
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  5. /2023 · 31 Jan · Shift 1 · Q45

Vector Algebra question

2023 · 31 Jan · Shift 1 · Q45

JEE MainMathematicsVector AlgebraNumerical+4 / −1
Let a⃗\vec{a}a and b⃗\vec{b}b be two vectors such that ∣a⃗∣=14,∣b⃗∣=6|\vec{a}|=\sqrt{14},|\vec{b}|=\sqrt{6}∣a∣=14​,∣b∣=6​ and ∣a⃗×b⃗∣=48|\vec{a} \times \vec{b}|=\sqrt{48}∣a×b∣=48​. Then (a⃗⋅b⃗)2(\vec{a} \cdot \vec{b})^{2}(a⋅b)2 is equal to ‾\underline{\hspace{2cm}}​.
Numerical answer
View written solutionFree

Correct answer: 36

  1. We use the identity relating dot product and cross product:

|\vec a ^2 |\vec b ^2 = (\vec a \cdot \vec b)^2 + |\vec a \times \vec b ^2

Equivalently,

(\vec a \cdot \vec b)^2 = |\vec a ^2 |\vec b ^2 - |\vec a \times \vec b ^2

  1. Substitute the given values:

|\vec a| = \sqrt{14} \Rightarrow |\vec a|^2 = 14 |\vec b| = \sqrt{6} \Rightarrow |\vec b|^2 = 6 |\vec a \times \vec b| = \sqrt{48} \Rightarrow |\vec a \times \vec b|^2 = 48

So,

(a⃗⋅b⃗)2=14⋅6−48(\vec a \cdot \vec b)^2 = 14 \cdot 6 - 48(a⋅b)2=14⋅6−48

  1. Compute:

14⋅6=8414 \cdot 6 = 8414⋅6=84 84−48=3684 - 48 = 3684−48=36

Hence,

(a⃗⋅b⃗)2=36(\vec a \cdot \vec b)^2 = 36(a⋅b)2=36

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