A vector r has magnitude 14 and direction ratios 2, 3, –6. Find the direction cosines and components of vector r,

Question

A vector \(\vec r\) has magnitude 14 and direction ratios 2, 3, –6. Find the direction cosines and components of \(\vec r\), given that \(\vec r\) makes an acute angle with x-axis.

Answer 1

Given that,

Magnitude of vector \(\vec r\) = 14

Know that, the direction cosines of a vector are the cosines of the angles between the vector and the three coordinate axes.

∴, the direction cosines l, m and n are

And we know that,

l² + m² + n² = 1

⇒ k² = 4

⇒ k = ±√4

⇒ k = ±2

Since, \(\vec r\) makes an acute angle with x-axis, then k will be positive.

⇒ k = 2

The direction cosines are

The components of \(\vec r\) can be found out by,