If vector p is a unit vector making a 45 angle with vector vector q and vector p· vector q=3sqrt(2) g find |vector q|

Let's solve the problem step by step.

Given:
- Vector \( \mathbf{p} \) is a unit vector.
- The angle between vector \( \mathbf{p} \) and vector \( \mathbf{q} \) is \( 45^\circ \).
- The dot product \( \mathbf{p} \cdot \mathbf{q} = 3\sqrt{2} \).

We need to find the magnitude of vector \( \mathbf{q} \), denoted as \( |\mathbf{q}| \).

### Step 1: Use the Dot Product Formula
The dot product of two vectors \( \mathbf{p} \) and \( \mathbf{q} \) is given by:
\[
\mathbf{p} \cdot \mathbf{q} = |\mathbf{p}| |\mathbf{q}| \cos \theta
\]
where \( \theta \) is the angle between the vectors.

### Step 2: Substitute the Given Values
Since \( \mathbf{p} \) is a unit vector, \( |\mathbf{p}| = 1 \). The angle \( \theta = 45^\circ \), so:
\[
\cos 45^\circ = \frac{1}{\sqrt{2}}
\]
Substituting these into the dot product formula:
\[
\mathbf{p} \cdot \mathbf{q} = 1 \times |\mathbf{q}| \times \frac{1}{\sqrt{2}}
\]

### Step 3: Set Up the Equation
We know that \( \mathbf{p} \cdot \mathbf{q} = 3\sqrt{2} \), so:
\[
3\sqrt{2} = |\mathbf{q}| \times \frac{1}{\sqrt{2}}
\]

### Step 4: Solve for \( |\mathbf{q}| \)
To solve for \( |\mathbf{q}| \), multiply both sides by \( \sqrt{2} \):
\[
|\mathbf{q}| = 3\sqrt{2} \times \sqrt{2}
\]
\[
|\mathbf{q}| = 3 \times 2 = 6
\]

### Final Answer:
The magnitude of vector \( \mathbf{q} \) is \( \boxed{6} \).

Would you like more details on any part of the solution?

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Here are 5 related questions for further exploration:

1. What happens to the dot product if the angle between the two vectors is \( 90^\circ \)?
2. How does the magnitude of a vector affect its direction?
3. Can you derive the formula for the dot product from the law of cosines?
4. How would the solution change if \( \mathbf{p} \) was not a unit vector?
5. What is the physical interpretation of the dot product in physics?

**Tip:** Remember, when dealing with unit vectors, their magnitude is always 1, which simplifies many vector calculations!

Learn how to find the magnitude of vector q when given the dot product and angle with vector p. Step-by-step solution with trigonometric and vector algebra concepts. Suitable for high school students.

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