Question

If \(\rm \vec a\) and \(\rm \vec b\) are vectors such that \(\rm |\vec a| = 13, |\vec b| = 2\) and \(\rm \vec a \times \vec b= 3{\rm{\hat i}} + 4{\rm{\hat j}} + 12{\rm{\hat k}}\) then what is the acute angle between \(\rm \vec a\) and \(\rm \vec b\)?
1. 60°
2. 45°
3. 30°
4. 90°

Answer 1

Correct Answer - Option 3 : 30°

Concept:

If two vector \(\rm \vec a\) and \(\rm \vec b\) then \(\rm |\vec a × \vec b| = \rm |\vec a| |\vec b| \sin θ\)

Where θ is the angle between vector  \(\rm \vec a\) and \(\rm \vec b\).

Consider vector  \(\rm \vec A = {\rm{x\hat i}} + {\rm{y\hat j}} + {\rm{z\hat k}}\) then magnitude of \(\rm \vec A \) =  |A| = \(\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2} + {{\rm{z}}^2}} \)

 

Calculation:

Given that,

\(\rm |\vec a| = 13, |\vec b| = 2\)

\(\rm \vec a × \vec b= 3{\rm{\hat i}} + 4{\rm{\hat j}} + 12{\rm{\hat k}}\)

Taking Magnitude both sides, we get

⇒ \(\rm |\vec a × \vec b|= |3{\rm{\hat i}} + 4{\rm{\hat j}} + 12{\rm{\hat k}}|\)

⇒ \(\rm |\vec a| |\vec b| \sin θ = \sqrt {3^2 + 4^2 + 12^2}\)

⇒ 13 × 2 × sin θ = \(\sqrt {169}\)

⇒ 13 × 2 × sin θ = 13

⇒ sin θ = \(\frac 12\)

∴ θ = 30°