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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Vector Algebra

NDA 2023 · Mathematics · Vector Algebra

If ( a × b )^2+( a · b )^2=144 and | b |=4 , then what is the value...

If (a⃗×b⃗)2+(a⃗⋅b⃗)2=144(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=144 and ∣b⃗∣=4|\vec{b}|=4, then what is the value of ∣a⃗∣|\vec{a}| ?

Options

  1. A3
  2. B4
  3. C5
  4. D6

Correct answer

A. 3

Step-by-step solution

 Given, (a→×b→)2+(a→⋅b→)2=144⇒∣a→∣2∣b→∣2→sin⁡2θ+∣a→∣2∣b→∣2cos⁡2θ=144⇒∣a→∣2∣b→∣2=144∣a→∣2×16=144⇒∣a→∣=3\begin{aligned} & \text{ Given, }(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})^2+(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}})^2=144 \\ & \Rightarrow|\overrightarrow{\mathrm{a}}|^2|\overrightarrow{\mathrm{ b}}|^2 \rightarrow \sin ^2 \theta+|\overrightarrow{\mathrm{a}}|^2|\overrightarrow{\mathrm{ b}}|^2 \cos ^2 \theta=144 \\ & \Rightarrow|\overrightarrow{\mathrm{a}}|^2|\overrightarrow{\mathrm{ b}}|^2=144 \\ & |\overrightarrow{\mathrm{a}}|^2 \times 16=144 \Rightarrow|\overrightarrow{\mathrm{a}}|=3\end{aligned}
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