For each pair of vectors u and v with angle θ between them, sketch the resultant.


|u| = 12, |v| = 20, θ = 27°

Verified step by step guidance

1. Step 1: Understand that the resultant vector \( \mathbf{R} \) is the vector sum of \( \mathbf{u} \) and \( \mathbf{v} \).
2. Step 2: Use the formula for the magnitude of the resultant vector: \( |\mathbf{R}| = \sqrt{|\mathbf{u}|^2 + |\mathbf{v}|^2 + 2|\mathbf{u}||\mathbf{v}|\cos(\theta)} \).
3. Step 3: Substitute the given values into the formula: \( |\mathbf{u}| = 12 \), \( |\mathbf{v}| = 20 \), and \( \theta = 27^\circ \).
4. Step 4: Calculate \( \cos(27^\circ) \) using a calculator or trigonometric table.
5. Step 5: Plug the values into the formula and simplify to find the magnitude of the resultant vector \( |\mathbf{R}| \).

Key Concepts

Vector Addition

Vector addition involves combining two or more vectors to find a resultant vector. This is typically done using the head-to-tail method or by applying the parallelogram law. The magnitude and direction of the resultant vector depend on the magnitudes of the individual vectors and the angle between them.

Magnitude of a Vector

The magnitude of a vector represents its length and is a measure of how strong or large the vector is. It is denoted by the vertical bars around the vector symbol, such as |u| or |v|. In this question, the magnitudes of vectors u and v are given as 12 and 20, respectively, which are essential for calculating the resultant vector.

Angle Between Vectors

The angle between two vectors is crucial for determining the direction of the resultant vector. In this case, the angle θ is given as 27°, which affects how the vectors combine. The angle can be used in calculations involving the cosine and sine functions to find the components of the resultant vector.