We are given two vectors:

$\mathbf{v}_1 = \langle 4, 1, -1 \rangle \quad \text{and} \quad \mathbf{v}_2 = \langle 1, x, 0 \rangle$

We need to find the values of $x$ such that the angle between these vectors is 45°.

The formula for the cosine of the angle $\theta$ between two vectors $\mathbf{v}_1$ and $\mathbf{v}_2$ is:

$\cos(\theta) = \frac{\mathbf{v}_1 \cdot \mathbf{v}_2}{|\mathbf{v}_1| |\mathbf{v}_2|}$

Step 1: Dot product $\mathbf{v}_1 \cdot \mathbf{v}_2$

The dot product of two vectors $\mathbf{v}_1 = \langle a_1, b_1, c_1 \rangle$ and $\mathbf{v}_2 = \langle a_2, b_2, c_2 \rangle$ is:

$\mathbf{v}_1 \cdot \mathbf{v}_2 = a_1a_2 + b_1b_2 + c_1c_2$

For our vectors $\mathbf{v}_1 = \langle 4, 1, -1 \rangle$ and $\mathbf{v}_2 = \langle 1, x, 0 \rangle$, the dot product is:

$\mathbf{v}_1 \cdot \mathbf{v}_2 = 4(1) + 1(x) + (-1)(0) = 4 + x$

Step 2: Magnitudes $|\mathbf{v}_1|$ and $|\mathbf{v}_2|$

The magnitude of a vector $\mathbf{v} = \langle a, b, c \rangle$ is given by:

$|\mathbf{v}| = \sqrt{a^2 + b^2 + c^2}$

For $\mathbf{v}_1 = \langle 4, 1, -1 \rangle$:

$|\mathbf{v}_1| = \sqrt{4^2 + 1^2 + (-1)^2} = \sqrt{16 + 1 + 1} = \sqrt{18} = 3\sqrt{2}$

For $\mathbf{v}_2 = \langle 1, x, 0 \rangle$:

$|\mathbf{v}_2| = \sqrt{1^2 + x^2 + 0^2} = \sqrt{1 + x^2}$

Step 3: Applying the cosine formula

We are given that the angle $\theta$ between the vectors is 45°, so:

$\cos(45^\circ) = \frac{\sqrt{2}}{2}$

Now, using the cosine formula:

$\frac{4 + x}{|\mathbf{v}_1| |\mathbf{v}_2|} = \frac{\sqrt{2}}{2}$

Substitute the magnitudes:

$\frac{4 + x}{(3\sqrt{2}) \cdot \sqrt{1 + x^2}} = \frac{\sqrt{2}}{2}$

Step 4: Solve for $x$

First, multiply both sides of the equation by $(3\sqrt{2}) \cdot \sqrt{1 + x^2}$:

$4 + x = \frac{\sqrt{2}}{2} \cdot 3\sqrt{2} \cdot \sqrt{1 + x^2}$

Simplifying the right-hand side:

$4 + x = 3 \cdot \sqrt{1 + x^2}$

Now, square both sides to eliminate the square root:

$(4 + x)^2 = 9(1 + x^2)$

Expand both sides:

$16 + 8x + x^2 = 9 + 9x^2$

Move all terms to one side:

$16 + 8x + x^2 - 9 - 9x^2 = 0$

Simplify:

$7 + 8x - 8x^2 = 0$

This is a quadratic equation. Rearranging:

$8x^2 - 8x - 7 = 0$

Step 5: Solve the quadratic equation

Use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For $8x^2 - 8x - 7 = 0$, $a = 8$, $b = -8$, and $c = -7$. Plug these values into the quadratic formula:

$x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(8)(-7)}}{2(8)}$

$x = \frac{8 \pm \sqrt{64 + 224}}{16}$

$x = \frac{8 \pm \sqrt{288}}{16}$

$x = \frac{8 \pm 12\sqrt{2}}{16}$

$x = \frac{2 \pm 3\sqrt{2}}{4}$

Thus, the two possible values for $x$ are:

$x = \frac{2 + 3\sqrt{2}}{4} \quad \text{or} \quad x = \frac{2 - 3\sqrt{2}}{4}$

Final Answer:

The values of $x$ are:

$x = \frac{2 + 3\sqrt{2}}{4}, \frac{2 - 3\sqrt{2}}{4}$

Would you like to see more details or have any questions on this?

Here are some related questions for you:

1. How do you calculate the angle between two vectors?
2. What is the geometric interpretation of the dot product?
3. How do you simplify square roots when solving equations?
4. What happens if the cosine of the angle is 0 or 1?
5. How do you solve a quadratic equation using the quadratic formula?

Tip: When solving for angles between vectors, always ensure that you calculate the magnitudes of the vectors correctly.