Question

Let \(\overset{⃗}{a}\) and \(\overset{⃗}{b}\) be linearly independent vectors such that
\(|\overset{⃗}{a}| = \sqrt{3},|\overset{⃗}{b}| = 3\) and \(|\overset{⃗}{a}-\overset{⃗}{b}| = 4\).
If \(\overset{⃗}{a}\times (2\hat{i}+2\hat{j}-\hat{k}) = (2\hat{i}+2\hat{j}-\hat{k})\times \overset{⃗}{b}\) and \(|(\overset{⃗}{a}+\overset{⃗}{b})\cdot (3\hat{i}+4\hat{j}+2\hat{k})| = \sqrt{λ}\), then \(λ = \ldots\)

• A. \(32\)
• B. \(64\)
• C. \(256\)
• D. \(128\)

Hint:

Use a + b + c = 0 to get b.c from the magnitudes.

Answer 1

The Correct Option is D

Step 1: Geometric picture:
Three vectors of equal length summing to zero form an equilateral triangle, so the angle between any two (tail to tail) is $120^\circ$.

Step 2: Evaluate:
$\tan120^\circ = -\sqrt3$, so $\tan^2 = 3$, and $\cot^2 = \frac13$. Sum $\frac{10}{3}$ (D).

Final Answer:
$\frac{10}{3}$. \[ \boxed{\frac{10}{3}} \]