In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 6, θ = 30°

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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 \( ||\mathbf{v}|| = 6 \) and the direction angle \( \theta = 30^\circ \) to find the components. The x-component is given by \( v_x = ||\mathbf{v}|| \cos \theta \), so write \( v_x = 6 \cos 30^\circ \).

Similarly, the y-component is given by \( v_y = ||\mathbf{v}|| \sin \theta \), so write \( v_y = 6 \sin 30^\circ \).

Substitute the values of \( v_x \) and \( v_y \) into the vector expression to write \( \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} \).

This gives the vector \( \mathbf{v} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \) using the given magnitude and direction angle.

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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 horizontal and vertical components, which correspond to the vector's projection on these axes.

Magnitude and Direction Angle of a Vector

The magnitude of a vector represents its length, while the direction angle θ indicates the angle it makes with the positive x-axis. These two parameters uniquely define the vector's orientation and size in the plane.

Using Trigonometry to Find Vector Components

The horizontal (x) and vertical (y) components of a vector can be found using trigonometric functions: x = ||v|| cos θ and y = ||v|| sin θ. This method converts polar form (magnitude and angle) into rectangular form (i and j components).