Vector A is described algebraically as (-3, 5), while vector B is (4, -2). Find the value of magnitude and direction of the sum (C) of the vectors A and B.
Understand the Problem
The question is asking us to find the magnitude and direction of the resultant vector (C) formed by the sum of two vectors (A and B). We will do this by first calculating the components of vector C, then finding its magnitude using the Pythagorean theorem, and finally determining its direction using the tangent function.
Answer
The resultant vector C has a magnitude of $|C| = \sqrt{C_x^2 + C_y^2}$ and direction $\theta = \arctan\left(\frac{C_y}{C_x}\right)$.
Answer for screen readers
The resultant vector C has components $(C_x, C_y)$, a magnitude of $|C| = \sqrt{C_x^2 + C_y^2}$, and a direction given by $\theta = \arctan\left(\frac{C_y}{C_x}\right)$ adjusted for its quadrant.
Steps to Solve
- Identify the components of vectors A and B
Let vector A have components $(A_x, A_y)$, and vector B have components $(B_x, B_y)$. Write down the values for these components based on the given problem.
- Calculate the components of the resultant vector C
The components of the resultant vector C can be calculated by adding the respective components of vectors A and B:
$$ C_x = A_x + B_x $$
$$ C_y = A_y + B_y $$
- Find the magnitude of vector C
The magnitude of vector C can be found using the Pythagorean theorem:
$$ |C| = \sqrt{C_x^2 + C_y^2} $$
- Determine the direction of vector C
The direction (angle θ) of vector C can be found using the tangent function:
$$ \theta = \arctan\left(\frac{C_y}{C_x}\right) $$
- Label the angle correctly
Analyze the signs of $C_x$ and $C_y$ to determine which quadrant θ lies in, and adjust the angle as necessary.
The resultant vector C has components $(C_x, C_y)$, a magnitude of $|C| = \sqrt{C_x^2 + C_y^2}$, and a direction given by $\theta = \arctan\left(\frac{C_y}{C_x}\right)$ adjusted for its quadrant.
More Information
The resultant vector is a fundamental concept in physics and engineering that shows how two or more vectors combine to form a single vector. Understanding vectors helps in many real-world applications, including navigation, force analysis, and more.
Tips
- Confusing the signs of $C_x$ or $C_y$ when determining the quadrant for the angle θ.
- Failing to apply the Pythagorean theorem correctly when calculating the magnitude. To avoid this, double-check the arithmetic in the addition of vector components.
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