Problem 7 - Algebra and Trigonometry

Question

Multiple Choice The angle \(\theta, 0 \leq \theta \leq \pi,\) between two nonzero vectors \(\mathbf{u}\) and \(\mathbf{v}\) can be found using what formula? (a) \(\sin \theta=\frac{\|\mathbf{u}\|}{\|\mathbf{v}\|}\) (b) \(\cos \theta=\frac{\|\mathbf{u}\|}{\|\mathbf{v}\|}\) (c) \(\sin \theta=\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}\) (d) \(\cos \theta=\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}\)

Step-by-Step Solution

Answer

(d) \(\text{cos} \theta = \frac{\text{u} \cdot \ \text{v}}{\|\textbf{u} \| \|\textbf{v} \|}\) is the correct formula.

Step 1: Recall the dot product formula

The dot product of two vectors \(\textbf{u}\) and \(\textbf{v}\) is given by: \(\textbf{u} \cdot \ \textbf{v} = \|\textbf{u}\|\textbf{v}\| \cos \theta\), where \( \theta\) is the angle between the vectors.

Step 2: Isolate the cosine term

From the dot product formula, isolating \( \cos \theta\), we get \(\text{cos} \theta = \frac{\text{u} \cdot \text{v}}{\|\textbf{u} \|\|\textbf{v} \|}\). This formula will help find the angle \( \theta\).

Step 3: Verify the choices

Match the correct formula from the options given. (a) is incorrect because it uses the sine of the angle, (b) is incorrect because it uses the cosine but the formula is inverted, and (c) is incorrect because it uses the sine of the angle. (d) is correct: \(\text{cos}\theta = \frac{\text{u} \cdot \text{v}}{\|\textbf{u} \|\|\textbf{v} \|}\).

Key Concepts

vector angle

To find the angle between two vectors, we can use information about their magnitudes and dot product. The angle between two vectors reveals how much they point in the same or opposite directions.
Vectors that point in the same direction have an angle of 0 degrees between them, while vectors that point in exactly opposite directions have an angle of 180 degrees (or \(\theta = \pi\) radians). And when vectors are perpendicular (at right angles), the angle between them is 90 degrees (or \(\theta = \frac{\pi}{2}\) radians).

The key to finding this angle is using the dot product of the vectors and understanding the relationship it has with the vectors' magnitudes. This brings us to the cosine formula, which is derived from the dot product formula.

cosine formula

The cosine formula is an essential part of understanding the relationship between vectors and their angles. The dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is given by:

\(\textbf{u} \cdot \textbf{v} = \|\textbf{u}\textbf{v}\textbf{v}\| \cos \theta\)

The angle \(\theta\) between them can be isolated in this formula:

\(\text{cos} \theta = \frac{\textbf{u} \cdot \textbf{v}}{\|\textbf{u}\|\|\textbf{v}\|}\)

This formula shows that the cosine of the angle \(\theta\) is equal to the dot product of the vectors \(\mathbf{u}\) and \(\mathbf{v}\) divided by the product of their magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\). This relationship is consistent across both 2D and 3D vectors.

Using this formula allows us to compute the angle, making it not only easier but also systematic to find the angle between any two given vectors.

trigonometry

Trigonometry is the branch of mathematics dealing with the relationships between the sides and angles of triangles. With vectors, it often involves the use of sine, cosine, and tangent functions, which relate the angles of a triangle to the lengths of its sides.

In the context of vectors, these trigonometric functions can help simplify and solve for angles between vectors. For example, in our exercise, the cosine function directly relates to the angle between two vectors via the cosine formula:

\(\text{cos} \theta = \frac{\textbf{u} \cdot \textbf{v}}{\|\textbf{u}\|\|\textbf{v}\|}\).

To recall, for any right triangle, the sine function refers to the ratio of the length of the opposite side to the hypotenuse, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side. In vector problems:

• The cosine function helps us find the angle between two vectors.
• The sine function isn't usually used to find angles directly in vector problems.

Trigonometry also helps in understanding concepts like projection, vector magnitude, and direction, critical in solving vector-related problems in various fields such as physics, engineering, and computer science.