Problem 8 - Algebra and Trigonometry

Question

Multiple Choice If two nonzero vectors \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal, then the angle between them has what measure? (a) \(\pi\) (b) \(\frac{\pi}{2}\) (c) \(\frac{3 \pi}{2}\) (d) \(2 \pi\)

Step-by-Step Solution

Answer

(b) \(\frac{\pi}{2}\)

Step 1: Understand the Definition of Orthogonal Vectors

Two vectors \(\textbf{v}\) and \(\textbf{w}\) are orthogonal if their dot product is zero. Mathematically, this can be written as \(\textbf{v} \cdot \textbf{w} = 0\). When the dot product is zero, it means the angle between the vectors is 90 degrees or \(\frac{\pi}{2}\) radians.

Step 2: Recall the Relationship Between Dot Product and Angle

The dot product of two vectors \(\textbf{v}\) and \(\textbf{w}\) is given by \[ \textbf{v} \cdot \textbf{w} = \|\textbf{v}\| \|\textbf{w}\| \cos(\theta) \] where \(\theta\) is the angle between the vectors, and \(\textbf{v}\) and \(\textbf{w}\) are non-zero vectors.

Step 3: Determine the Angle Based on Orthogonality

Since \(\textbf{v} \cdot \textbf{w} = 0\), it follows that \( \cos(\theta) = 0 \). The cosine of which angle is zero? \(\cos(\frac{\pi}{2}) = 0\). Therefore, the angle \(\theta\) between orthogonal vectors is \(\frac{\pi}{2}\) radians.

Step 4: Choose the Correct Option

Given the options: \((a) \pi, (b) \frac{\pi}{2}, (c) \frac{3 \pi}{2}, (d) 2 \pi\), the correct measure of the angle between two orthogonal vectors is \(\frac{\pi}{2}\). Therefore, the correct answer is (b).

Key Concepts

dot product

To understand orthogonal vectors, we first need to know what the dot product is. The dot product (or scalar product) between two vectors, \(\textbf{v}\) and \(\textbf{w}\), is a measure of how similar the two vectors are in direction. The dot product is given by the formula: \[ \textbf{v} \cdot \textbf{w} = \|\textbf{v}\| \|\textbf{w}\| \cos(\theta)\] where \(\theta\) is the angle between the vectors. In simpler terms, this formula multiplies the lengths (or magnitudes) of the vectors and the cosine of the angle between them. \ When two vectors are orthogonal, which means they are at a right angle to each other, their dot product is zero. For example, if \(\textbf{v}\) and \(\textbf{w}\) are orthogonal, \(\textbf{v} \cdot \textbf{w} = 0\). This is a crucial property to keep in mind. It tells us that there is no 'directional similarity' between orthogonal vectors. They are completely independent in their directions.

angle between vectors

Knowing the angle between two vectors is essential to understanding their relationship. The angle \(\theta\) between two vectors can be found using the dot product formula mentioned earlier: \[ \cos(\theta) = \frac{\textbf{v}\cdot\textbf{w}}{\|\textbf{v}\||\textbf{w}\|}\text{.}\] When vectors are orthogonal, \(\textbf{v} \cdot \textbf{w}\) equals zero, which simplifies our calculations. With the dot product being zero, we need to find the angle whose cosine value is zero. The angle that satisfies this is 90 degrees, or \(\frac{\pi}{2}\) radians. This demonstrates that orthogonal vectors always meet at a right angle. \ Here are a few other key angles (in degrees and radians) that you should be familiar with when dealing with vectors: \

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• 0 degrees = 0 radians (vectors point in the same direction)
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• 90 degrees = \frac{\pi}{2} radians (orthogonal vectors)
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• 180 degrees = \pi radians (vectors point in opposite directions)
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Getting comfortable with these angles will make it easier to visualize and work with vector relationships.

cosine function

The cosine function is a crucial part of understanding vector relationships, especially when considering the angle between vectors. Cosine, abbreviated as \(\cos\), is a trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the length of the hypotenuse. In the context of vectors, it helps us determine how 'aligned' two vectors are. \ The cosine of 90 degrees (or \(\frac{\pi}{2}\) radians) is zero. This is why, when two vectors are orthogonal, their dot product equals zero. They form a right angle, making the cosine of the angle between them zero. \ To clarify: \

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• When \(\theta\) is 0 degrees, \(\cos(\theta) = 1\). This means the vectors are pointing in the same direction.
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• When \(\theta\) is 90 degrees (orthogonal vectors), \(\cos(\theta) = 0\).
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• When \(\theta\) is 180 degrees, \(\cos(\theta) = -1\). This indicates the vectors are pointing in exactly opposite directions.
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Understanding the cosine function and its values for key angles is fundamental in studying vectors and their properties.