Question

Find the angle between the following pairs of vectors. a. \(\mathbf{u}=\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right], \mathbf{v}=\left[\begin{array}{l}2 \\ 0 \\ 1\end{array}\right]\) b. \(\mathbf{u}=\left[\begin{array}{r}3 \\ -1 \\ 0\end{array}\right], \mathbf{v}=\left[\begin{array}{r}-6 \\ 2 \\ 0\end{array}\right]\) c. \(\mathbf{u}=\left[\begin{array}{r}7 \\ -1 \\ 3\end{array}\right], \mathbf{v}=\left[\begin{array}{r}1 \\ 4 \\ -1\end{array}\right]\) d. \(\mathbf{u}=\left[\begin{array}{r}2 \\ 1 \\ -1\end{array}\right], \mathbf{v}=\left[\begin{array}{l}3 \\ 6 \\ 3\end{array}\right]\) e. \(\mathbf{u}=\left[\begin{array}{r}1 \\ -1 \\ 0\end{array}\right], \mathbf{v}=\left[\begin{array}{l}0 \\ 1 \\ 1\end{array}\right]\) f. \(\mathbf{u}=\left[\begin{array}{l}0 \\ 3 \\ 4\end{array}\right], \mathbf{v}=\left[\begin{array}{r}5 \sqrt{2} \\ -7 \\ -1\end{array}\right]\)

Short answer

Angle between vectors (a) is 33.69°, (b) is 0°, (c) is 83.90°, (d) is 38.21°, (e) is 90°, (f) is 87.95°.

Step-by-step solution

Understand the Formula

The formula to find the angle \( \theta \) between two vectors \( \mathbf{u} = [u_1, u_2, u_3] \) and \( \mathbf{v} = [v_1, v_2, v_3] \) is given by: \[\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}\] where \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \) is the dot product, and \( \|\mathbf{u}\| \) and \( \|\mathbf{v}\| \) are the magnitudes of the vectors, calculated as \( \|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2 + u_3^2} \) and \( \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2} \).

Key concepts

Dot Product

The dot product is a fundamental operation for calculating the angle between two vectors. It provides a way to multiply vectors that results in a scalar. The dot product of two vectors, \(\mathbf{u} = [u_1, u_2, u_3]\) and \(\mathbf{v} = [v_1, v_2, v_3]\), is calculated as:\[ \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \] This formula essentially sums up the products of the corresponding components of the vectors.
Key points to remember about the dot product:

• It results in a scalar, not another vector.
• It's commutative, which means \(\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}\).
• If the dot product equals zero, the vectors are perpendicular.

Understanding the dot product is crucial for further calculations like finding angles between vectors.

Vector Magnitude

Magnitude of a vector is its length and is always a positive value. To find the magnitude of a vector, such as \( \mathbf{u} = [u_1, u_2, u_3]\), you use the formula:\[ \| \mathbf{u} \| = \sqrt{u_1^2 + u_2^2 + u_3^2} \] This uses the components of the vector squared, summed, and then square-rooted.
Some essentials about vector magnitude:

• The formula shows a Pythagorean approach, extending the idea of line lengths to multiple dimensions.
• Magnitudes are always non-negative numbers.
• A zero magnitude indicates a zero vector, meaning all components are zero.

Magnitude is vital for normalization and calculating angles, giving the vector's size without regard to direction.

Cosine Formula for Vectors

The angle \( \theta \) between two vectors can be determined using the cosine formula. This relates the dot product and the magnitudes of the vectors. The formula is as follows:\[ \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \] Breaking it down:

• The numerator is the dot product, \( \mathbf{u} \cdot \mathbf{v}\), telling us about the directional alignment of vectors.
• The denominator combines the magnitudes of the vectors, normalizing the dot product to give a ratio.
• The resulting value from \(\cos \theta\) is typically between -1 and 1, interpreted via the cosine function to find the angle in radians or degrees.

Understanding this formula helps in determining whether vectors are pointed in the same direction, opposite, or are perpendicular.