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

Given that,

Magnitude of vector = 14

Also, direction ratios = 2 : 3 : -6

Also can be defined as,

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

⇒ 49k² = 196

⇒ k² = 4

⇒ k = ±√4

⇒ k = ±2

Since, makes an acute angle with x-axis, then k will be positive.

⇒ k = 2

The direction cosines are

The components of can be found out by,

Thus, the direction cosines (l, m, n) are ; and the components of are .