Let's solve each part of the problem step by step.

Part (a)

Given:

• Initial point of vector $\mathbf{u}$ is $(-5, -1)$
• Terminal point of vector $\mathbf{u}$ is $(x, 0)$
• The vector $\mathbf{u}$ points in the same direction as $\mathbf{v} = (1, 3)$

To find the components of vector $\mathbf{u}$, we first express it as: $\mathbf{u} = (x - (-5), 0 - (-1)) = (x + 5, 1)$ This vector must be a scalar multiple of $\mathbf{v} = (1, 3)$, so: $\mathbf{u} = k(1, 3) = (k, 3k)$ Therefore, we have the equations: $x + 5 = k \quad \text{and} \quad 1 = 3k$

Solving for $k$ in the second equation: $k = \frac{1}{3}$

Now, substitute $k = \frac{1}{3}$ into the first equation: $x + 5 = \frac{1}{3}$

Solving for $x$: $x = \frac{1}{3} - 5 = \frac{1}{3} - \frac{15}{3} = \frac{-14}{3}$

So the vector $\mathbf{u}$ is: $\mathbf{u} = \left(\frac{1}{3}, 1\right)$

Part (b)

Given:

• Initial point of vector $\mathbf{w}$ is $(0, y)$
• Terminal point of vector $\mathbf{w}$ is $(4, 2)$
• The vector $\mathbf{w}$ points in the same direction as $\mathbf{v} = (3, 1)$

To find the components of vector $\mathbf{w}$, we express it as: $\mathbf{w} = (4 - 0, 2 - y) = (4, 2 - y)$ This vector must be a scalar multiple of $\mathbf{v} = (3, 1)$, so: $\mathbf{w} = k(3, 1) = (3k, k)$ Therefore, we have the equations: $4 = 3k \quad \text{and} \quad 2 - y = k$

Solving for $k$ in the first equation: $k = \frac{4}{3}$

Substitute $k = \frac{4}{3}$ into the second equation: $2 - y = \frac{4}{3}$

Solving for $y$: $y = 2 - \frac{4}{3} = \frac{6}{3} - \frac{4}{3} = \frac{2}{3}$

So the vector $\mathbf{w}$ is: $\mathbf{w} = \left(4, \frac{2}{3}\right)$

Summary of Results

• Part (a): The components of vector $\mathbf{u}$ are $\left(\frac{1}{3}, 1\right)$.
• Part (b): The components of vector $\mathbf{w}$ are $\left(4, \frac{2}{3}\right)$.

Would you like more details or have any questions? Here are some related questions to consider:

1. How do you determine the direction of a vector?
2. What is the significance of scalar multiplication in vectors?
3. How would you find the magnitude of the vectors $\mathbf{u}$ and $\mathbf{w}$?
4. What happens if the initial or terminal points of the vector are modified?
5. How do you normalize a vector, and why might it be useful?

Tip: When solving vector problems, always check if the direction and magnitude make sense in the context of the problem.