Find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the nearest tenth of a degree, for each given vector v.


v = (7i - 3j) - (10i - 3j)

Verified step by step guidance

1. Combine the vector components: $v=(7i-3j)-(10i-3j)$.
2. Simplify the vector by subtracting the components: $v=(7-10)i+(-3+3)j$.
3. This results in the vector $v=-3i+0j$.
4. Calculate the magnitude of the vector using the formula $||v||=\sqrt{(-3{)}^2+0^2}$.
5. Determine the direction angle $\theta$ using $\theta ={\tan }^{-1}\left(\frac{0}{-3}\right)$, and adjust for the correct quadrant.

Key Concepts

Vector Subtraction

Vector subtraction involves finding the difference between two vectors by subtracting their corresponding components. For the vectors v = (7i - 3j) and (10i - 3j), the subtraction is performed component-wise, resulting in a new vector that represents the change in position from one vector to the other.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the formula ||v|| = √(x² + y²), where x and y are the components of the vector. In this case, after finding the resultant vector from the subtraction, the magnitude can be computed to determine how far the vector extends in space.

Direction Angle of a Vector

The direction angle θ of a vector is the angle formed between the vector and the positive x-axis, typically measured in degrees. It can be calculated using the tangent function, θ = arctan(y/x), where y and x are the components of the vector. This angle provides insight into the orientation of the vector in the Cartesian plane.