Write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given.
||v|| = 10, θ = 330°
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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 \( ||\mathbf{v}|| = 10 \) and the direction angle \( \theta = 330^\circ \) to find the components. The formulas for the components are: \[ v_x = ||\mathbf{v}|| \cos(\theta) \quad \text{and} \quad v_y = ||\mathbf{v}|| \sin(\theta) \]
Substitute the given values into the component formulas: \[ v_x = 10 \cos(330^\circ) \quad \text{and} \quad v_y = 10 \sin(330^\circ) \]
Evaluate the trigonometric functions \( \cos(330^\circ) \) and \( \sin(330^\circ) \) using the unit circle or known values for standard angles. Remember that \( 330^\circ = 360^\circ - 30^\circ \), so you can use reference angles to find these values.
Write the vector \( \mathbf{v} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \) by plugging in the calculated components: \[ \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} \]
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 (x) and vertical (y) components, which fully describe its direction and magnitude.
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 allow us to determine the vector’s components using trigonometric functions.
The x-component of a vector is found by multiplying its magnitude by cos(θ), and the y-component by multiplying the magnitude by sin(θ). This method converts polar form (magnitude and angle) into rectangular form (i and j components).