Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction. (d) a = (20i + 10j) m/s^2

Verified step by step guidance

1. Represent the vector a = (20i + 10j) m/s^2 on a coordinate system. Draw the vector starting from the origin (0,0) and ending at the point (20,10).
2. Label the angle \( \theta \) between the vector and the positive x-axis. This angle can be found using the tangent function, where \( \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \).
3. Calculate the magnitude of the vector using the Pythagorean theorem. The magnitude (or length) of the vector a is given by \( \|a\| = \sqrt{a_x^2 + a_y^2} \), where \( a_x \) and \( a_y \) are the components of the vector along the x and y axes, respectively.
4. Determine the direction \( \theta \) by calculating \( \theta = \tan^{-1}\left(\frac{a_y}{a_x}\right) \).
5. Summarize the magnitude and direction of the vector. The magnitude is the result from step 3, and the direction is the angle \( \theta \) calculated in step 4, measured in degrees from the positive x-axis.

Key Concepts

Vector Representation

Vectors are quantities that have both magnitude and direction, represented in a coordinate system. In this case, the vector a = (20i + 10j) m/s² can be visualized in a two-dimensional plane, where 'i' represents the x-component and 'j' represents the y-component. Understanding how to draw and label vectors is crucial for visualizing their direction and magnitude.

Magnitude of a Vector

The magnitude of a vector is a measure of its length, calculated using the Pythagorean theorem. For the vector a = (20i + 10j) m/s², the magnitude can be found using the formula |a| = √(x² + y²), where x and y are the components of the vector. This results in |a| = √(20² + 10²) m/s², which quantifies the overall strength of the vector.

Direction of a Vector

The direction of a vector is specified by the angle it makes with a reference axis, typically the positive x-axis. This angle can be calculated using the tangent function, where θ = arctan(y/x). For the vector a = (20i + 10j) m/s², the angle can be determined by finding θ = arctan(10/20), which provides insight into how the vector is oriented in the coordinate system.