To find the angle $\theta$ between two vectors $\mathbf{u}$ and $\mathbf{v}$, we use the following formula:

$\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}$

Where:

• $\mathbf{u} \cdot \mathbf{v}$ is the dot product of the two vectors,
• $\|\mathbf{u}\|$ is the magnitude of $\mathbf{u}$,
• $\|\mathbf{v}\|$ is the magnitude of $\mathbf{v}$.

Given: $\mathbf{u} = -4j + 6k$ $\mathbf{v} = -3i + j + 5k$

Step 1: Compute the dot product $\mathbf{u} \cdot \mathbf{v}$

The dot product of two vectors is given by: $\mathbf{u} \cdot \mathbf{v} = (u_i \cdot v_i) + (u_j \cdot v_j) + (u_k \cdot v_k)$

Since $\mathbf{u} = 0i - 4j + 6k$ and $\mathbf{v} = -3i + j + 5k$, their dot product is: $\mathbf{u} \cdot \mathbf{v} = (0 \cdot -3) + (-4 \cdot 1) + (6 \cdot 5)$ $\mathbf{u} \cdot \mathbf{v} = 0 - 4 + 30 = 26$

Step 2: Compute the magnitudes of $\mathbf{u}$ and $\mathbf{v}$

The magnitude of a vector $\mathbf{a} = a_i i + a_j j + a_k k$ is: $\|\mathbf{a}\| = \sqrt{a_i^2 + a_j^2 + a_k^2}$

For $\mathbf{u} = 0i - 4j + 6k$: $\|\mathbf{u}\| = \sqrt{0^2 + (-4)^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52}$

For $\mathbf{v} = -3i + j + 5k$: $\|\mathbf{v}\| = \sqrt{(-3)^2 + 1^2 + 5^2} = \sqrt{9 + 1 + 25} = \sqrt{35}$

Step 3: Compute the cosine of the angle

Now, substitute the values into the formula: $\cos \theta = \frac{26}{\sqrt{52} \cdot \sqrt{35}} = \frac{26}{\sqrt{1820}}$

$\cos \theta = \frac{26}{42.68} \approx 0.609$

Step 4: Find the angle $\theta$

To find $\theta$, take the inverse cosine (arccos): $\theta = \cos^{-1}(0.609) \approx 52.7^\circ$

Thus, the angle between the two vectors is approximately $52.7^\circ$.

Would you like more details or have any questions?

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Here are 5 related questions:

1. How is the dot product of two vectors used in physics?
2. What is the significance of the magnitude of a vector?
3. How do you calculate the angle between vectors in higher dimensions?
4. What happens if the dot product of two vectors is zero?
5. How does the cross product differ from the dot product?

Tip: The dot product gives the projection of one vector onto another, helping you find angles between vectors.