To determine the magnitude of the vector $\mathbf{a} + \mathbf{b}$, we need to first add the two vectors and then calculate the magnitude of the resulting vector.

Given: $\mathbf{a} = \begin{pmatrix} 1 \\ -3 \end{pmatrix}$ $\mathbf{b} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}$

1. Adding the vectors: $\mathbf{a} + \mathbf{b} = \begin{pmatrix} 1 + 5 \\ -3 + 2 \end{pmatrix} = \begin{pmatrix} 6 \\ -1 \end{pmatrix}$

2. Calculating the magnitude of $\mathbf{a} + \mathbf{b}$: $|\mathbf{a} + \mathbf{b}| = \sqrt{6^2 + (-1)^2} = \sqrt{36 + 1} = \sqrt{37}$

Thus, the magnitude of $\mathbf{a} + \mathbf{b}$ is $\sqrt{37}$.

Answer

The correct answer is: $\sqrt{37}$, which corresponds to Option 3: 37.

Would you like further explanation, or do you have any questions about this solution?

Related Questions

1. How do you calculate the magnitude of a vector in any dimension?
2. What is the significance of vector addition in physics and engineering applications?
3. How does the dot product of two vectors differ from vector addition?
4. Can two vectors of different dimensions be added? Why or why not?
5. How does the magnitude of a vector change if both components are doubled?

Tip

Remember, the magnitude of a vector $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$ is calculated as $\sqrt{x^2 + y^2}$. This is essentially the distance formula applied to the origin and the point represented by the vector.