Problem 2 - Algebra and Trigonometry

Question

Let \(\mathbf{a}=\left\langle a_{1}, a_{2}\right\rangle\) and \(\mathbf{b}=\left\langle b_{1}, b_{2}\right\rangle\) be nonzero vectors in the plane, and let \(\theta\) be the angle between them. The angle \(\theta\) satisfies \(\cos \theta=\) _____ So if \(\mathbf{a} \cdot \mathbf{b}=0,\) the vectors are _____

Step-by-Step Solution

Answer

Orthogonal, if \(\mathbf{a} \cdot \mathbf{b}=0\). \(\cos \theta=\frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|\|\mathbf{b}\|}\).

Step 1: Recall the Formula for the Dot Product and Its Relation to Angle

The dot product of two vectors \( \mathbf{a} = \left\langle a_1, a_2 \right\rangle \) and \( \mathbf{b} = \left\langle b_1, b_2 \right\rangle \) is given by \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \). Additionally, the dot product can also be expressed in terms of the angle \( \theta \) between the vectors: \( \mathbf{a} \cdot \mathbf{b} = \| \mathbf{a} \| \| \mathbf{b} \| \cos \theta \), where \( \| \mathbf{a} \| \) and \( \| \mathbf{b} \| \) are the magnitudes of vectors \( \mathbf{a} \) and \( \mathbf{b} \) respectively.

Step 2: Express the Cosine of the Angle

From the relationship \( \mathbf{a} \cdot \mathbf{b} = \| \mathbf{a} \| \| \mathbf{b} \| \cos \theta \), we can express the cosine of the angle between the vectors as \[\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \| \| \mathbf{b} \|}.\] This expression shows that the cosine of the angle depends on the dot product and the magnitudes of the vectors.

Step 3: Condition When Dot Product is Zero

When \( \mathbf{a} \cdot \mathbf{b} = 0 \), according to the dot product formula, the numerator of the cosine expression becomes zero:\[\cos \theta = \frac{0}{\| \mathbf{a} \| \| \mathbf{b} \|} = 0,\]which implies that \( \theta = \frac{\pi}{2} \). Thus, if the dot product is zero, the angle \( \theta \) between the vectors is \( 90^\circ \), meaning the vectors are orthogonal.

Key Concepts

Dot Product

The dot product is a fundamental concept in vector mathematics. It's a way of multiplying two vectors that results in a scalar. For two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), the dot product, symbolized as \( \mathbf{a} \cdot \mathbf{b} \), is calculated by:

• Multiplying the corresponding components: \( a_1b_1 + a_2b_2 \).
• Summing these products.

This formula is essential for finding scalar quantities from vector operations. Unlike vector multiplication in physics, the result isn't a vector but a single number. The dot product also forms the basis for understanding vector projections and angles between vectors.

Angle Between Vectors

Understanding the angle between vectors involves using the dot product and vector magnitudes. The angle \( \theta \) between two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is characterized by how aligned they are. Using the dot product, the angle can be determined through the formula:\[\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \| \| \mathbf{b} \|} \]Here:

• \( \| \mathbf{a} \| \) and \( \| \mathbf{b} \| \) are the magnitudes of the vectors.
• \( \cos \theta \) describes the cosine of the angle between the vectors.

This equation reveals that when the dot product is positive, the angle is acute, meaning less than \( 90^\circ \), and when it is negative, the angle is obtuse, more than \( 90^\circ \). The formula helps us recognize how two vectors interact in space through their orientation.

Orthogonal Vectors

Vectors are orthogonal when they meet at a right angle. Mathematically, this condition is indicated by the dot product being zero: \( \mathbf{a} \cdot \mathbf{b} = 0 \). If this happens, the angle \( \theta \) between them is \( 90^\circ \).What implies orthogonality?

• The vectors have no directional overlap.
• Their relationship in space is purely perpendicular.

Understanding orthogonality is crucial in many fields like geometry, physics, and computer graphics. Perpendicular vectors often simplify problems due to their lack of directional influence on each other. Recognizing orthogonal vectors involves checking for a zero dot product, confirming they have a right-angle relationship.