The magnitude and direction angle of v are ||v|| = 12 and θ = 60°. Express v in terms of i and j.

검증된 단계별 안내

Recall that a vector \( \mathbf{v} \) in two dimensions 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 \( \mathbf{v} \) along the x-axis and y-axis respectively.

Use the magnitude \( ||\mathbf{v}|| = 12 \) and the direction angle \( \theta = 60^\circ \) to find the components. The x-component is given by \( v_x = ||\mathbf{v}|| \cos \theta \).

Similarly, the y-component is given by \( v_y = ||\mathbf{v}|| \sin \theta \).

Substitute the known values into the component formulas: \( v_x = 12 \cos 60^\circ \) and \( v_y = 12 \sin 60^\circ \).

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

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주요 개념

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

Vector Magnitude and Direction

A vector's magnitude represents its length or size, while the direction angle indicates the angle it makes with the positive x-axis. Together, these define the vector's position in the plane, allowing conversion between polar and rectangular forms.

Unit Vectors i and j

Unit vectors i and j represent the standard basis vectors along the x-axis and y-axis, respectively. Expressing a vector in terms of i and j means breaking it down into horizontal and vertical components.

Conversion from Polar to Rectangular Coordinates

To express a vector given by magnitude and angle in terms of i and j, use trigonometric functions: the x-component is magnitude times cosine of the angle, and the y-component is magnitude times sine of the angle.