Problem 24 - University Calculus: Early Transcendentals

Question

Let \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}\) \(\mathbf{v}=-\mathbf{i}+\mathbf{j}+\mathbf{k}, \quad \mathbf{w}=\mathbf{i}+\mathbf{k}, \quad \mathbf{r}=-(\pi / 2) \mathbf{i}-\pi \mathbf{j}+(\pi / 2) \mathbf{k}\) Which vectors, if any, are (a) perpendicular? (b) Parallel? Give reasons for your answers.

Step-by-Step Solution

Answer

Vectors \( \mathbf{u} \) and \( \mathbf{v} \), \( \mathbf{u} \) and \( \mathbf{w} \), \( \mathbf{v} \) and \( \mathbf{w} \), \( \mathbf{v} \) and \( \mathbf{r} \), \( \mathbf{w} \) and \( \mathbf{r} \) are perpendicular; no vectors are parallel.

Step 1: Review Vector Definitions

In this exercise, we need to analyze and compare the given vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w}, \mathbf{r} \). They are expressed in terms of their components along the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \).

Step 2: Define Perpendicular Vectors Condition

Two vectors are perpendicular if their dot product is zero. Recall that the dot product of two vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \) is given by: \[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]

Step 3: Calculate Dot Products to Find Perpendicular Vectors

For each pair of vectors, calculate the dot product to determine if they're perpendicular: 1. \( \mathbf{u} \cdot \mathbf{v} = (1)(-1) + (2)(1) + (-1)(1) = -1 + 2 - 1 = 0 \) 2. \( \mathbf{u} \cdot \mathbf{w} = (1)(1) + (2)(0) + (-1)(1) = 1 + 0 - 1 = 0 \) 3. \( \mathbf{u} \cdot \mathbf{r} = (1)(-\pi/2) + (2)(-\pi) + (-1)(\pi/2) = -\pi/2 - 2\pi - \pi/2 = -3\pi \) 4. \( \mathbf{v} \cdot \mathbf{w} = (-1)(1) + (1)(0) + (1)(1) = -1 + 0 + 1 = 0 \) 5. \( \mathbf{v} \cdot \mathbf{r} = (-1)(-\pi/2) + (1)(-\pi) + (1)(\pi/2) = \pi/2 - \pi + \pi/2 = 0 \) 6. \( \mathbf{w} \cdot \mathbf{r} = (1)(-\pi/2) + (0)(-\pi) + (1)(\pi/2) = -\pi/2 + \pi/2 = 0 \)

Step 4: Identify Perpendicular Vectors

From the dot product calculations in Step 3, observe that the following pairs have a dot product of zero and are therefore perpendicular: - \( \mathbf{u} \) and \( \mathbf{v} \) - \( \mathbf{u} \) and \( \mathbf{w} \) - \( \mathbf{v} \) and \( \mathbf{w} \) - \( \mathbf{v} \) and \( \mathbf{r} \) - \( \mathbf{w} \) and \( \mathbf{r} \)

Step 5: Define Parallel Vectors Condition

Two vectors are parallel if they are scalar multiples of each other. This means \( \mathbf{a} = k \mathbf{b} \) for a scalar \( k \).

Step 6: Check for Parallel Vectors

Examine each pair of vectors: - For \( \mathbf{u} \) and \( \mathbf{v} \), check if one is a scalar multiple of the other. They are not. - For \( \mathbf{u} \) and \( \mathbf{w} \), not scalar multiples. - For \( \mathbf{u} \) and \( \mathbf{r} \), not scalar multiples. - For \( \mathbf{v} \) and \( \mathbf{w} \), not scalar multiples. - For \( \mathbf{v} \) and \( \mathbf{r} \), not scalar multiples. - For \( \mathbf{w} \) and \( \mathbf{r} \), not scalar multiples. Therefore, no vectors are parallel.

Key Concepts

Perpendicular Vectors

In vector analysis, perpendicular vectors have a special property: their dot product equals zero. The dot product of two vectors measures how much one vector goes in the direction of another. It's calculated using their components:\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]If this result is zero, the vectors are perpendicular, meaning they form a right angle (90 degrees) with each other.

• To find out which vectors are perpendicular in an exercise, calculate the dot product for each pair.

In our example, vectors \( \mathbf{u} \) and \( \mathbf{v} \) are perpendicular because their dot product is zero, as are the pairs \( \mathbf{u} \) and \( \mathbf{w} \), \( \mathbf{v} \) and \( \mathbf{w} \), among others. This concept helps in understanding how multi-dimensional paths relate visually and conceptually.

Parallel Vectors

Parallel vectors share the same directionality, even if they vary in magnitude. They are aligned and differ only by a scalar. This means one vector is a scaled version of another:\[ \mathbf{a} = k \mathbf{b} \]

• "\( k \)" is the scalar multiple defining how much one vector is stretched or compressed.

When vectors are parallel, they appear to be overlapping if viewed from the direction they are pointing. If you suspect two vectors might be parallel, check if one is a simple multiple of the other using their components. If you cannot find such a scalar, they are not parallel. In the provided exercise, none of the vectors are parallel to each other. Recognizing parallel vectors is crucial in many fields, where proportional relationships between vectors are analyzed.

Dot Product

The dot product is an operation that takes two vectors and returns a single number, a scalar, which gives important information about the vectors' interaction.\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]

• When the dot product is zero, the vectors are perpendicular.
• If the dot product is positive, vectors point in more or less the same direction.
• A negative dot product suggests the vectors point in opposite directions.

The dot product is not only useful for determining perpendicular vectors but also for finding angles between vectors and their projections. It is a building block for more advanced vector operations and has broad applications, such as in physics for calculating work done along a force direction.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a single number). This operation scales the vector's magnitude without changing its direction.

• If \( c \) is a scalar and \( \mathbf{a} \) is a vector, then \( c \mathbf{a} \) is a new vector obtained by multiplicating each component of \( \mathbf{a} \) by \( c \).

For example, if \( \mathbf{a} = 3\mathbf{i} + 4\mathbf{j} \) and we multiply it by 2, the resulting vector would be \( 6\mathbf{i} + 8\mathbf{j} \). Scalar multiplication is fundamental in determining parallel vectors and simplifies the description of vector relationships where proportional scaling is necessary, such as in graphical transformations and physics.