To find the resultant magnitude of a vector and the angles it makes with the coordinate axes ($\alpha$, $\beta$, and $\gamma$), you can follow these steps:

Step 1: Sum the Components

Given a vector in 3D space with components along the $i$, $j$, and $k$ directions:

$\vec{A} = A_i \hat{i} + A_j \hat{j} + A_k \hat{k}$

The resultant magnitude ($|\vec{A}|$) of the vector is given by:

$|\vec{A}| = \sqrt{A_i^2 + A_j^2 + A_k^2}$

Step 2: Find Angles with the Axes

The angles $\alpha$, $\beta$, and $\gamma$ represent the angles between the vector and the $x$, $y$, and $z$ axes, respectively. These angles can be found using the direction cosines:

• $\cos(\alpha) = \frac{A_i}{|\vec{A}|}$
• $\cos(\beta) = \frac{A_j}{|\vec{A}|}$
• $\cos(\gamma) = \frac{A_k}{|\vec{A}|}$

From these, the angles can be calculated as:

• $\alpha = \cos^{-1}\left(\frac{A_i}{|\vec{A}|}\right)$
• $\beta = \cos^{-1}\left(\frac{A_j}{|\vec{A}|}\right)$
• $\gamma = \cos^{-1}\left(\frac{A_k}{|\vec{A}|}\right)$

Would you like to go through a specific example or have any questions?

Here are five related questions you might consider:

1. How do you determine the direction of the resultant vector in 3D space?
2. What happens to the angles $\alpha$, $\beta$, and $\gamma$ if the vector is along one of the axes?
3. How do you interpret negative values for vector components in relation to their angles?
4. Can a vector have the same magnitude but different angles with respect to the axes?
5. How do vector projections relate to the calculation of these angles?

Tip: Always make sure your vector components are in the correct units before calculating the resultant magnitude and angles to avoid errors in the results.