Let's solve this problem step by step.

Given:

• Vector A has a magnitude of 10.6 units and points due west.
• Vector B points due north.
• The magnitude of the vector sum A + B is 19.6 units.

Part (a): Magnitude of B if A + B has a magnitude of 19.6 units

We know that A and B are perpendicular to each other because A points west and B points north. Therefore, we can use the Pythagorean theorem to find the magnitude of B:

$\text{Magnitude of } A + B = \sqrt{A^2 + B^2}$

Substituting the given values:

$19.6 = \sqrt{(10.6)^2 + B^2}$

Now, solving for B:

$19.6^2 = (10.6)^2 + B^2$ $384.16 = 112.36 + B^2$ $B^2 = 384.16 - 112.36$ $B^2 = 271.8$ $B = \sqrt{271.8} \approx 16.48 \text{ units}$

So, the magnitude of B is approximately 16.48 units.

Part (b): Direction of A + B relative to due west

To find the direction of the vector A + B relative to due west, we can use the tangent function:

$\tan(\theta) = \frac{B}{A} = \frac{16.48}{10.6}$

Calculating the angle θ:

$\theta = \tan^{-1}\left(\frac{16.48}{10.6}\right) \approx \tan^{-1}(1.554) \approx 57.5^\circ \text{ north of west}$

So, the direction of A + B relative to due west is approximately 57.5° north of west.

Part (c): Magnitude of B if A - B has a magnitude of 19.6 units

For the vector difference A - B, we again use the Pythagorean theorem:

$\text{Magnitude of } A - B = \sqrt{A^2 + B^2}$

Since A - B is also perpendicular (but pointing in different directions), the magnitude is the same as A + B:

$19.6 = \sqrt{(10.6)^2 + B^2}$

We already solved this in part (a), so the magnitude of B is again 16.48 units.

Part (d): Direction of A - B relative to due west

For the direction of A - B, we calculate:

$\tan(\theta) = \frac{B}{A} = \frac{16.48}{10.6}$

But this time, the direction is south of west since B is subtracted:

$\theta = \tan^{-1}\left(\frac{16.48}{10.6}\right) \approx 57.5^\circ \text{ south of west}$

So, the direction of A - B relative to due west is approximately 57.5° south of west.

Summary of Results:

(a) Magnitude of B = 16.48 units

(b) Direction of A + B = 57.5° north of west

(c) Magnitude of B (for A - B) = 16.48 units

(d) Direction of A - B = 57.5° south of west

Would you like any further details or explanations on these results?

Here are some related questions for further exploration:

1. How would the results change if vector B pointed east instead of north?
2. What would be the magnitude and direction of A - 2B?
3. How do these vector operations apply to real-world navigation?
4. Can you calculate the resultant vector if B had a different magnitude?
5. How does changing the angle between A and B affect the results?

Tip: When working with vector problems, always visualize the vectors on a coordinate system to help understand their directions and relationships better.