To find the projection of b onto a, denoted as $\text{proj}_{\mathbf{a}} \mathbf{b}$, we use the formula:

$\text{proj}_{\mathbf{a}} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a}$

Step 1: Calculate $\mathbf{a} \cdot \mathbf{b}$

The dot product $\mathbf{a} \cdot \mathbf{b}$ is calculated as follows: $\mathbf{a} \cdot \mathbf{b} = (-5)(3) + (-1)(-1) + (1)(-2) = -15 + 1 - 2 = -16$

Step 2: Calculate $\mathbf{a} \cdot \mathbf{a}$

The dot product $\mathbf{a} \cdot \mathbf{a}$ is: $\mathbf{a} \cdot \mathbf{a} = (-5)^2 + (-1)^2 + (1)^2 = 25 + 1 + 1 = 27$

Step 3: Substitute into the formula

Now we can find $\text{proj}_{\mathbf{a}} \mathbf{b}$ by substituting these values: $\text{proj}_{\mathbf{a}} \mathbf{b} = \frac{-16}{27} \mathbf{a} = \frac{-16}{27} \langle -5, -1, 1 \rangle$

Step 4: Distribute the scalar

Now, multiply each component of $\mathbf{a}$ by $\frac{-16}{27}$: $\text{proj}_{\mathbf{a}} \mathbf{b} = \left\langle \frac{-16}{27} \cdot (-5), \frac{-16}{27} \cdot (-1), \frac{-16}{27} \cdot 1 \right\rangle$

Calculating each component:

1. $\frac{-16}{27} \cdot (-5) = \frac{80}{27}$
2. $\frac{-16}{27} \cdot (-1) = \frac{16}{27}$
3. $\frac{-16}{27} \cdot 1 = \frac{-16}{27}$

Final Answer

$\text{proj}_{\mathbf{a}} \mathbf{b} = \left\langle \frac{80}{27}, \frac{16}{27}, \frac{-16}{27} \right\rangle$

Let me know if you have questions or want further details.

Related Questions

1. How do you calculate the angle between two vectors using the dot product?
2. What is the difference between the projection of b onto a and a onto b?
3. Can projections result in a vector with larger magnitude than the original vector?
4. How would you find the magnitude of the projection vector?
5. How do you interpret a negative result in the dot product of two vectors?

Tip

The projection of one vector onto another always lies along the direction of the vector it's being projected onto.