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. First, simplify the given vector expression by subtracting the components of the vectors: \(v = (7\mathbf{i} - 3\mathbf{j}) - (10\mathbf{i} - 3\mathbf{j})\). This means subtract the \(i\) components and the \(j\) components separately.
2. Calculate the resulting vector components: \(v = (7 - 10)\mathbf{i} + (-3 - (-3))\mathbf{j}\). Simplify these to find the components of \(v\).
3. Find the magnitude \(||v||\) of the vector using the formula \(||v|| = \sqrt{v_x^2 + v_y^2}\), where \(v_x\) and \(v_y\) are the \(i\) and \(j\) components of the vector respectively.
4. Determine the direction angle \(\theta\) of the vector relative to the positive \(x\)-axis using the formula \(\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right)\). Make sure to consider the quadrant in which the vector lies to find the correct angle.
5. Convert the angle \(\theta\) from radians to degrees if necessary, and round the magnitude to the nearest hundredth and the angle to the nearest tenth of a degree as requested.

Key Concepts

Vector Subtraction

Vector subtraction involves subtracting corresponding components of two vectors. For vectors in component form, subtract the i-components and j-components separately to find the resultant vector. This operation is essential to simplify the given vector expression before further calculations.

Magnitude of a Vector

The magnitude of a vector v = (x, y) is the length of the vector, calculated using the Pythagorean theorem as ||v|| = √(x² + y²). This scalar value represents the distance from the origin to the point (x, y) in the plane.

Direction Angle of a Vector

The direction angle θ of a vector is the angle it makes with the positive x-axis, found using θ = arctangent(y/x). It is usually measured in degrees and indicates the vector's orientation in the plane, requiring adjustment based on the vector's quadrant.