Multiple Choice
Given the line with parametric equations , , , find the perpendicular distance from the point to the line.
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8. Vectors
Geometric Vectors
Problem 19
Textbook Question
For each pair of vectors u and v with angle θ between them, sketch the resultant.
|u| = 12, |v| = 20, θ = 27°
Verified step by step guidance1
Identify the given information: the magnitudes of vectors \( \mathbf{u} \) and \( \mathbf{v} \) are \( |\mathbf{u}| = 12 \) and \( |\mathbf{v}| = 20 \), and the angle between them is \( \theta = 27^\circ \).
Recall that the resultant vector \( \mathbf{R} = \mathbf{u} + \mathbf{v} \) can be found using the Law of Cosines for vectors, where the magnitude of \( \mathbf{R} \) is given by:
\[
|\mathbf{R}| = \sqrt{|\mathbf{u}|^2 + |\mathbf{v}|^2 + 2 |\mathbf{u}| |\mathbf{v}| \cos(\theta)}
\]
To sketch the resultant, start by drawing vector \( \mathbf{u} \) with length proportional to 12 units in any direction you choose.
Next, from the tip of \( \mathbf{u} \), draw vector \( \mathbf{v} \) at an angle of \( 27^\circ \) relative to \( \mathbf{u} \), with length proportional to 20 units.
The resultant vector \( \mathbf{R} \) is then drawn from the tail of \( \mathbf{u} \) (the starting point) to the tip of \( \mathbf{v} \) (the ending point). This vector represents the sum \( \mathbf{u} + \mathbf{v} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Addition
Vector addition involves combining two vectors to find their resultant by placing them head-to-tail and drawing the vector from the start of the first to the end of the second. This graphical method helps visualize the resultant vector's magnitude and direction.
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Adding Vectors Geometrically
Law of Cosines for Vectors
The Law of Cosines relates the magnitudes of two vectors and the angle between them to find the magnitude of their resultant: |R| = √(|u|² + |v|² + 2|u||v|cosθ). This formula is essential for calculating the exact length of the resultant vector.
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Angle Between Vectors
The angle θ between two vectors determines how they combine. It affects both the magnitude and direction of the resultant vector, influencing whether the vectors reinforce or partially cancel each other.
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