The image shows a
vector addition problem where two vectors,
A and
B, are being added to find the
resultant vector R. Let's go through the solution step by step and explain it clearly.
---
🔷 Problem Statement:
Determine the
resultant vector R of:
-
A = 30.0 N at 0° (along the positive x-axis)
-
B = 40.0 N at 90° (along the positive y-axis)
---
🔷 Step 1: Understand the Vectors
- Vector
A is horizontal (east direction), magnitude = 30.0 N.
- Vector
B is vertical (north direction), magnitude = 40.0 N.
- These two vectors are
perpendicular to each other (angle between them = 90°).
So, when you add them, they form a
right triangle, with the resultant
R as the hypotenuse.
---
🔷 Step 2: Use Pythagorean Theorem to Find Magnitude of Resultant
We use:
$$
|R| = \sqrt{A^2 + B^2}
$$
Substitute values:
$$
|R| = \sqrt{(30.0)^2 + (40.0)^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50.0\ \text{N}
$$
✔ So, the
magnitude of the resultant is
50.0 N.
---
🔷 Step 3: Find the Direction of the Resultant (Angle)
We use trigonometry to find the angle $ x $ that the resultant makes with the x-axis (i.e., with vector A).
$$
\tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{B}{A} = \frac{40.0}{30.0} = 1.33
$$
Now take the inverse tangent:
$$
x = \tan^{-1}(1.33) \approx 53^\circ
$$
✔ So, the
direction of the resultant is
53° above the positive x-axis.
---
🔷 Step 4: Express the Resultant
Since vector A was at 0° and we measured the angle from there, the bearing (or direction) of the resultant is:
$$
\text{Bearing} = 0^\circ + 53^\circ = 53^\circ
$$
---
✔ Final Answer:
$$
\boxed{R = 50\ \text{N at } 53^\circ}
$$
This means the resultant force has a magnitude of
50 Newtons and is directed
53 degrees north of east.
---
📌 Summary of Key Concepts Used:
-
Pythagorean Theorem: For perpendicular vectors, $ R = \sqrt{A^2 + B^2} $
-
Tangent function: To find the angle, $ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} $
-
Resultant vector: Combines magnitude and direction of two or more vectors.
---
💡 Visual Interpretation:
The diagram shows:
- Vector A pointing right (→),
- Vector B pointing up (↑),
- Resultant R (red arrow) diagonally upward from the origin,
- Angle $ x = 53^\circ $ between R and A.
All calculations match this geometric configuration.
---
✔ Answer is correct:
$$
\boxed{R = 50\ \text{N at } 53^\circ}
$$
Parent Tip: Review the logic above to help your child master the concept of vectors worksheet answers.
