A vector acts at the origin in a direction defined by the angles ∅y=65° and ∅z=40°. Knowing that the component of the x-axis vector is 7, determine the missing components and draw the angle ex

To find the missing components of the vector, we start by using the direction cosines and the given component along the x-axis:

Given:
- Component along x-axis, V_x = 7
- Direction cosines: m = \cos(65°), n = \cos(40°)

1. Find l using the equation l^2 + m^2 + n^2 = 1:
l = \sqrt{1 - m^2 - n^2} = \sqrt{1 - \cos^2(65°) - \cos^2(40°)}

2. Calculate l:
l = \sqrt{1 - \cos^2(65°) - \cos^2(40°)} \approx \sqrt{1 - 0.4226 - 0.7660} \approx 0.201

3. Find the magnitude of the vector, V:
V = \frac{V_x}{l} = \frac{7}{0.201} \approx 34.83

4. Calculate the components along y and z axes:
- V_y = V \cdot m \approx 34.83 \cdot \cos(65°) \approx 6.11
- V_z = V \cdot n \approx 34.83 \cdot \cos(40°) \approx 11.07

Therefore, the missing components of the vector are:
- V_y \approx 6.11
- V_z \approx 11.07

The angle \phi_x between the vector and the x-axis is approximately 61.03°.

\boxed{V_y \approx 6.11, V_z \approx 11.07, \phi_x \approx 61.03°}