Direction Cosines Calculation

To find the direction cosines of a vector, we need to divide each component of the vector by its magnitude.

Vector Components and Magnitude

The given vector is $\vec{v} = 3i - 2j + 6k$. The components of the vector are:

• $v_x = 3$
• $v_y = -2$
• $v_z = 6$

First, calculate the magnitude of the vector, $||\vec{v}||$. The magnitude is calculated using the formula:

$ ||\vec{v}|| = \sqrt{v_x^2 + v_y^2 + v_z^2} $

Substitute the component values:

$ ||\vec{v}|| = \sqrt{(3)^2 + (-2)^2 + (6)^2} $

$ ||\vec{v}|| = \sqrt{9 + 4 + 36} $

$ ||\vec{v}|| = \sqrt{49} $

$ ||\vec{v}|| = 7 $

Determining Direction Cosines

The direction cosines ($l, m, n$) are found using the formulas:

• $l = \frac{v_x}{||\vec{v}||}$
• $m = \frac{v_y}{||\vec{v}||}$
• $n = \frac{v_z}{||\vec{v}||}$

Now, substitute the component values and the calculated magnitude:

• $l = \frac{3}{7}$
• $m = \frac{-2}{7}$
• $n = \frac{6}{7}$

Therefore, the direction cosines of the vector $3i - 2j + 6k$ are $[3/7, -2/7, 6/7]$.