Textbook Question
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 4i - 2j
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8. Vectors
Unit Vectors and i & j Notation
2:38 minutesProblem 49
Textbook Question
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°
Verified step by step guidance1
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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Video duration:
2mKey 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 components, which represent its horizontal and vertical displacements.
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Introduction to Vectors
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.
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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.
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