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|| = 6, θ = 30°
8. Vectors
Geometric Vectors
2:57 minutesProblem 41
Textbook Question
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 3i - 4j
Verified step by step guidance1
Identify the given vector \( \mathbf{v} = 3\mathbf{i} - 4\mathbf{j} \). This means the vector components are \( v_x = 3 \) and \( v_y = -4 \).
Calculate the magnitude (length) of the vector \( \mathbf{v} \) using the formula:
\[ \text{magnitude} = ||\mathbf{v}|| = \sqrt{v_x^2 + v_y^2} \]
Substitute the components to get:
\[ ||\mathbf{v}|| = \sqrt{3^2 + (-4)^2} \]
Simplify the expression inside the square root to find the magnitude, but do not calculate the final numeric value yet.
To find the unit vector \( \mathbf{u} \) in the same direction as \( \mathbf{v} \), divide each component of \( \mathbf{v} \) by its magnitude:
\[ \mathbf{u} = \left( \frac{v_x}{||\mathbf{v}||}, \frac{v_y}{||\mathbf{v}||} \right) \]
Write the unit vector explicitly as:
\[ \mathbf{u} = \frac{1}{||\mathbf{v}||} (3\mathbf{i} - 4\mathbf{j}) \]
This expresses the unit vector in the same direction as \( \mathbf{v} \).
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Components and Notation
A vector in two dimensions is expressed in terms of its components along the x and y axes, often written as v = ai + bj, where i and j are unit vectors in the x and y directions. Understanding this notation helps in identifying the vector's direction and magnitude.
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i & j Notation
Magnitude of a Vector
The magnitude (or length) of a vector v = ai + bj is found using the Pythagorean theorem: |v| = √(a² + b²). This scalar value represents the distance from the origin to the point (a, b) and is essential for normalizing the vector.
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Unit Vector and Normalization
A unit vector has a magnitude of 1 and points in the same direction as the original vector. To find it, divide each component of the vector by its magnitude, resulting in a vector of length one that preserves the original direction.
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