A pilot heads a jet due east. The jet has a speed of 400 mi/h relative to the air. The wind is blowing due north with a speed of 30 mi/h. (Assume that the i vector points east, and the j vector points north.)
(a)
Express the velocity of the wind as a vector in component form.

(b)
Express the velocity of the jet relative to the air as a vector in component form.

(c)
Find the true velocity of the jet as a vector.

(d)
Find the true speed of the jet (in mi/h). (Round your answer to the nearest integer.)
mi/h
Find the direction of the jet (in degrees E of N). (Round your answer to one decimal place.)
N
° E

Let's go through each part of this problem step by step.

### Given Data:
- Jet speed relative to the air = 400 mi/h due east.
- Wind speed = 30 mi/h due north.
  
We'll define vectors in terms of \( \mathbf{i} \) (east) and \( \mathbf{j} \) (north).

### (a) Velocity of the Wind as a Vector
The wind is blowing due north at 30 mi/h. Therefore, the wind vector has no component in the \( \mathbf{i} \) (east) direction and is entirely in the \( \mathbf{j} \) (north) direction.

\[
\mathbf{v}_{\text{wind}} = 0 \mathbf{i} + 30 \mathbf{j} = \begin{pmatrix} 0 \\ 30 \end{pmatrix}
\]

### (b) Velocity of the Jet Relative to the Air as a Vector
The jet is heading due east at 400 mi/h relative to the air. This means the jet's velocity relative to the air has no north component and is entirely in the east direction.

\[
\mathbf{v}_{\text{jet, air}} = 400 \mathbf{i} + 0 \mathbf{j} = \begin{pmatrix} 400 \\ 0 \end{pmatrix}
\]

### (c) True Velocity of the Jet as a Vector
The true velocity of the jet, \( \mathbf{v}_{\text{true}} \), is the vector sum of the jet's velocity relative to the air and the wind's velocity.

\[
\mathbf{v}_{\text{true}} = \mathbf{v}_{\text{jet, air}} + \mathbf{v}_{\text{wind}}
\]
\[
\mathbf{v}_{\text{true}} = \begin{pmatrix} 400 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 30 \end{pmatrix} = \begin{pmatrix} 400 \\ 30 \end{pmatrix}
\]

### (d) True Speed of the Jet
The true speed of the jet is the magnitude of the true velocity vector.

\[
\text{True Speed} = \sqrt{(400)^2 + (30)^2}
\]
\[
= \sqrt{160000 + 900} = \sqrt{160900} \approx 401 \text{ mi/h}
\]

### Direction of the Jet
To find the direction of the jet, we calculate the angle \( \theta \) east of north. This is the angle formed by the true velocity vector with respect to the north direction.

\[
\theta = \tan^{-1} \left( \frac{400}{30} \right)
\]
\[
\theta = \tan^{-1} \left( \frac{400}{30} \right) \approx \tan^{-1}(13.33) \approx 85.7^\circ
\]

### Summary of Answers
- **(a)** Velocity of the wind as a vector: \( \begin{pmatrix} 0 \\ 30 \end{pmatrix} \)
- **(b)** Velocity of the jet relative to the air as a vector: \( \begin{pmatrix} 400 \\ 0 \end{pmatrix} \)
- **(c)** True velocity of the jet as a vector: \( \begin{pmatrix} 400 \\ 30 \end{pmatrix} \)
- **(d)** True speed of the jet: 401 mi/h
- **Direction of the jet**: \( 85.7^\circ \) E of N

Would you like more details on any specific part of this solution?

---

1. How would the calculations change if the wind were blowing from the south?
2. What would happen to the jet’s true velocity if the wind speed increased?
3. How can we find the components of the jet’s velocity if the jet were heading northeast instead of east?
4. How does changing the direction of the i and j vectors (east/north) impact vector calculations?
5. How can we calculate the direction if the angle was given relative to the east?

**Tip:** When working with vector problems, break down each motion into component vectors; this simplifies combining vectors significantly!

A pilot heads a jet due east. The jet has a speed of 400 mi/h relative to the air. The wind is blowing due north with a speed of 30 mi/h. (Assume that the i vector points east, and the j vector points north.)
(a) Express the velocity of the wind as a vector in component form.
(b) Express the velocity of the jet relative to the air as a vector in component form.
(c) Find the true velocity of the jet as a vector.
(d) Find the true speed of the jet (in mi/h). (Round your answer to the nearest integer.) mi/h
Find the direction of the jet (in degrees E of N). (Round your answer to one decimal place.) N ° E.

Solve for the true velocity and direction of a jet moving at 400 mi/h eastward relative to the air, with a wind blowing 30 mi/h northward. This problem covers vector addition, vector components, and trigonometric functions to determine speed and angle. Suitable for advanced high school math.

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