Given 

Bold v equals negative 3 Bold i minus Bold j**v**=−3**i**−**j** 

and Bold w equals negative Bold i minus 3 Bold j**w**=−**i**−3**j**

​(a) find the dot product 

Bold v times Bold w**v**•**w**​;

​(b) find the angle between 

Bold v**v** 

and 

Bold w**w**.




We are given the vectors:

\[
\mathbf{v} = -3\mathbf{i} - \mathbf{j}
\]
and 
\[
\mathbf{w} = -\mathbf{i} - 3\mathbf{j}.
\]

### (a) Find the dot product \(\mathbf{v} \cdot \mathbf{w}\):

The dot product of two vectors \(\mathbf{v} = a_1 \mathbf{i} + b_1 \mathbf{j}\) and \(\mathbf{w} = a_2 \mathbf{i} + b_2 \mathbf{j}\) is given by:
\[
\mathbf{v} \cdot \mathbf{w} = a_1 a_2 + b_1 b_2.
\]

For \(\mathbf{v} = -3\mathbf{i} - \mathbf{j}\), we have:
\[
a_1 = -3, \quad b_1 = -1.
\]

For \(\mathbf{w} = -\mathbf{i} - 3\mathbf{j}\), we have:
\[
a_2 = -1, \quad b_2 = -3.
\]

Now, compute the dot product:
\[
\mathbf{v} \cdot \mathbf{w} = (-3)(-1) + (-1)(-3) = 3 + 3 = 6.
\]

So, the dot product \(\mathbf{v} \cdot \mathbf{w} = 6\).

### (b) Find the angle \(\theta\) between \(\mathbf{v}\) and \(\mathbf{w}\):

The formula to find the angle between two vectors using the dot product is:
\[
\cos \theta = \frac{\mathbf{v} \cdot \mathbf{w}}{|\mathbf{v}| |\mathbf{w}|}.
\]

First, calculate the magnitudes of \(\mathbf{v}\) and \(\mathbf{w}\):

\[
|\mathbf{v}| = \sqrt{(-3)^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10},
\]
\[
|\mathbf{w}| = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}.
\]

Now, substitute the values into the formula for \(\cos \theta\):
\[
\cos \theta = \frac{6}{\sqrt{10} \cdot \sqrt{10}} = \frac{6}{10} = 0.6.
\]

Finally, take the inverse cosine to find the angle \(\theta\):
\[
\theta = \cos^{-1}(0.6).
\]

Using a calculator:
\[
\theta \approx 53.13^\circ.
\]

### Final answers:
(a) \(\mathbf{v} \cdot \mathbf{w} = 6\).

(b) The angle between \(\mathbf{v}\) and \(\mathbf{w}\) is approximately \(53.13^\circ\).

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the physical interpretation of the dot product?
2. How does the angle between vectors affect their dot product?
3. What happens if the dot product between two vectors is zero?
4. How would you compute the cross product of two vectors in 3D space?
5. Can two non-zero vectors ever have a dot product greater than their magnitudes?

**Tip:** The dot product can be used to determine if two vectors are perpendicular; if their dot product is zero, they are orthogonal.

Given **v** = -3**i** - **j** and **w** = -**i** - 3**j**, (a) find the dot product **v**•**w**; (b) find the angle between **v** and **w**.

This solution explains how to find the dot product of vectors v = -3i - j and w = -i - 3j and how to calculate the angle between them using vector algebra concepts. Suitable for students in Grades 10-12, this problem covers the dot product theorem and angle between vectors.

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