Add up the I j k componets and take the square root to find the resultant magnitude after finding that find alpha betta and gamma for the result net

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.

Learn how to find the magnitude and angles of a 3D vector using direction cosines and the law of cosines. This problem involves summing vector components and calculating angles with the coordinate axes. Suitable for advanced high school or college-level mathematics.

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