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

Verified step by step guidance

1. Step 1: Understand the vector components. The vector \( \mathbf{a} \) is given as \( \mathbf{a} = 20\mathbf{i} + 10\mathbf{j} \), where \( \mathbf{i} \) represents the x-component and \( \mathbf{j} \) represents the y-component. This means the x-component of the vector is 20 m/s², and the y-component is 10 m/s².
2. Step 2: Draw the vector. On a Cartesian coordinate system, plot the x-component (20 m/s²) along the positive x-axis and the y-component (10 m/s²) along the positive y-axis. The vector \( \mathbf{a} \) is represented as the diagonal of the rectangle formed by these components, starting from the origin.
3. Step 3: Calculate the magnitude of the vector. Use the Pythagorean theorem: \( |\mathbf{a}| = \sqrt{(a_x)^2 + (a_y)^2} \), where \( a_x = 20 \) m/s² and \( a_y = 10 \) m/s². Substitute these values into the formula to find the magnitude.
4. Step 4: Determine the direction of the vector. The direction is given by the angle \( \theta \) that the vector makes with the positive x-axis. Use the formula \( \theta = \tan^{-1}\left(\frac{a_y}{a_x}\right) \), where \( a_x = 20 \) m/s² and \( a_y = 10 \) m/s². Substitute these values to find the angle.
5. Step 5: Label the vector. On your diagram, label the vector \( \mathbf{a} \) with its magnitude and the angle \( \theta \) you calculated. Ensure the angle is measured counterclockwise 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 graphically represent vectors is essential 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 concept is crucial for determining how strong or large the vector is in physical terms.

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, finding the angle helps in understanding how the vector is oriented in space, which is important for applications in physics.