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

검증된 단계별 안내

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.

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.

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.

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.

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.