In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 12, θ = 225°

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Recall that a vector \( \mathbf{v} \) in the plane can be expressed in terms of the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) as \( \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} \), where \( v_x \) and \( v_y \) are the components of the vector along the x-axis and y-axis respectively.

Use the magnitude \( ||\mathbf{v}|| = 12 \) and the direction angle \( \theta = 225^\circ \) to find the components. The formulas for the components are: \[ v_x = ||\mathbf{v}|| \cos(\theta) \] \[ v_y = ||\mathbf{v}|| \sin(\theta) \]

Substitute the given values into the component formulas: \[ v_x = 12 \cos(225^\circ) \] \[ v_y = 12 \sin(225^\circ) \]

Evaluate the cosine and sine of \( 225^\circ \). Remember that \( 225^\circ \) is in the third quadrant where both sine and cosine are negative. Use the reference angle \( 225^\circ - 180^\circ = 45^\circ \) to find the exact values.

Write the vector \( \mathbf{v} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \) using the components found: \[ \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} \]

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주요 개념

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Vector Representation in the Plane

A vector in two dimensions can be expressed as a combination of unit vectors i and j along the x- and y-axes, respectively. Writing a vector in terms of i and j involves finding its components, which represent its horizontal and vertical displacements.

Magnitude and Direction Angle of a Vector

The magnitude of a vector is its length, denoted ||v||, and the direction angle θ is the angle it makes with the positive x-axis, measured counterclockwise. These two parameters uniquely define the vector's position in the plane.

Converting Polar Form to Cartesian Components

To express a vector given by magnitude and direction angle in terms of i and j, use trigonometric functions: the x-component is ||v|| cos θ and the y-component is ||v|| sin θ. This conversion translates polar coordinates into Cartesian vector components.