Question

Vector a points in the positive x direction and has a magnitude of 75 m. The vector c = a + b points in the positive y direction and has a magnitude of 95 m.

(a) Sketch the vectors a, b, and c.
(b) Estimate the magnitude and direction of the vector b from your sketch.

Answer 1

(a) Vector
`\( \mathbf{a} \)` with a magnitude of 75 m points in the positive x-direction. Vector
`\( \mathbf{c} = \mathbf{a} + \mathbf{b} \)` has a magnitude of 95 m and points in the positive y-direction. Vector
`\( \mathbf{b} \)` is perpendicular to a and has a magnitude of 60 m.

(b) The magnitude of vector b is 60 m, and its direction is in the positive z-direction.

Step-by-step explanation:

In the sketch, vector a is represented as an arrow of 75 units pointing in the positive x-direction. Vector c is the sum of vectors b and b and is shown as an arrow of 95 units pointing in the positive y-direction. Since vector
`\( \mathbf{b} \) is perpendicular to \( \mathbf{a} \)` and completes a right-angled triangle with
`\( \mathbf{a} \) and \( \mathbf{c} \),` we can use the Pythagorean theorem to find the magnitude of
`\( \mathbf{b} \).`

`\[ \|\mathbf{b}\| = \sqrt{\|\mathbf{c}\|^2 - \|\mathbf{a}\|^2} \]`

`\[ \|\mathbf{b}\| = √(95^2 - 75^2) = √(9025 - 5625) = √(3400) = 60 \, \text{m} \]`

The direction of b is determined by its orientation in the sketch. Since b is perpendicular to the plane formed by
`\( \mathbf{a} \) and \( \mathbf{c} \),` it points in the positive z-direction.

In conclusion, vector b has a magnitude of 60 m and points in the positive z-direction, completing the vector triangle formed by
`\( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \).`

Answer 2

The estimated magnitude of vector b is approximately 58.31 meters, and the estimated direction is approximately 51.88 degrees above the positive x-axis.

(a) To sketch the vectors a, b, and c:

1. Start with vector a, which points in the positive x direction and has a magnitude of 75 m. Draw an arrow to the right (positive x direction) and label it "a = 75 m."

2. Vector c is the result of adding vector a and vector b. Since vector c points in the positive y direction and has a magnitude of 95 m, draw an arrow upward (positive y direction) starting from the end of vector a. Label it "c = 95 m."

Now, we need to estimate the magnitude and direction of vector b.

(b) To estimate the magnitude and direction of vector b:

1. Draw vector b starting from the tail of vector a and ending at the head of vector c, completing the triangle.

2. To estimate the magnitude of vector b, you can use the Pythagorean theorem since you have a right triangle formed by vectors a, b, and c.

Magnitude of b (|b|) = √(Magnitude of c (|c|)² - Magnitude of a (|a|)²)

Magnitude of b (|b|) = √(95 m)² - (75 m)²)

Magnitude of b (|b|) = √(9025 m² - 5625 m²)

Magnitude of b (|b|) = √(3400 m²)

Magnitude of b (|b|) = 58.31 m (approximately)

3. To estimate the direction of vector b, you can use trigonometry. The angle θ between vectors b and c can be found using the tangent function:

tan(θ) = (Magnitude of a (|a|)) / (Magnitude of b (|b|))

tan(θ) = 75 m / 58.31 m

θ ≈ arctan(75 m / 58.31 m)

θ ≈ 51.88 degrees

So, the estimated magnitude of vector b is approximately 58.31 meters, and the estimated direction is approximately 51.88 degrees above the positive x-axis.