Problem 24 - Thomas Calculus

Question

Parallel and perpendicular vectors 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: Identify Conditions for Perpendicularity and Parallelism

Two vectors are said to be perpendicular if their dot product is zero. For 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 \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \). If this equals zero, the vectors are perpendicular. For parallel vectors, one is a scalar multiple of the other.

Step 2: Calculate Dot Product of \( \mathbf{u} \) and \( \mathbf{v} \)

Find the dot product \( \mathbf{u} \cdot \mathbf{v} \) where \( \mathbf{u} = 5 \mathbf{i} - \mathbf{j} + \mathbf{k} \) and \( \mathbf{v} = \mathbf{j} - 5 \mathbf{k} \). Compute:\[(5)(0) + (-1)(1) + (1)(-5) = 0 - 1 - 5 = -6\]Since \(-6 eq 0\), vectors \( \mathbf{u} \) and \( \mathbf{v} \) are not perpendicular.

Step 3: Calculate Dot Product of \( \mathbf{u} \) and \( \mathbf{w} \)

Find \( \mathbf{u} \cdot \mathbf{w} \) where \( \mathbf{w} = -15 \mathbf{i} + 3 \mathbf{j} - 3 \mathbf{k} \):\[(5)(-15) + (-1)(3) + (1)(-3) = -75 - 3 - 3 = -81\]Since \(-81 eq 0\), \( \mathbf{u} \) and \( \mathbf{w} \) are not perpendicular.

Step 4: Calculate Dot Product of \( \mathbf{v} \) and \( \mathbf{w} \)

Compute \( \mathbf{v} \cdot \mathbf{w} \):\[(0)(-15) + (1)(3) + (-5)(-3) = 0 + 3 + 15 = 18\]Since \(18 eq 0\), \( \mathbf{v} \) and \( \mathbf{w} \) are not perpendicular.

Step 5: Check for Parallelism

Check if any vector is a scalar multiple of another:1. Consider \( \mathbf{u} = 5 \mathbf{i} - \mathbf{j} + \mathbf{k} \) and \( \mathbf{v} = \mathbf{j} - 5 \mathbf{k} \): Not parallel because the \( \mathbf{i} \) component of \( \mathbf{v} \) is 0.2. Consider \( \mathbf{u} \) and \( \mathbf{w} = -15 \mathbf{i} + 3 \mathbf{j} - 3 \mathbf{k} \): Not parallel since there's no constant \( k \) where both vectors match in all components.3. Consider \( \mathbf{v} \) and \( \mathbf{w} \): Not parallel as components don't align even considering sign changes or ratios.

Key Concepts

Perpendicular Vectors

When we talk about vectors being perpendicular, we mean that they are at a right angle to each other, forming a perfect "L" shape. This characteristic is crucial when studying vector properties, as it shows how two directions can be completely independent of each other.

To determine if two vectors are perpendicular, we use the dot product of the two vectors. In mathematical terms, the dot product \[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]should equal zero. This implies that there's no overlap in direction between the two vectors at all.

• If the result is zero, the vectors are perpendicular.
• If not, they aren't perpendicular.

In the exercise, for example, we found that the dot products were not zero, so none of the given pairs of vectors were perpendicular.

Parallel Vectors

When vectors are parallel, they move in the same direction or in directly opposite directions. This can be imagined like two lanes on a straight highway.

For vectors to be parallel, one vector must be a scalar multiple of the other. This means that each component of one vector is multiplied by the same number to obtain the corresponding component in the other vector.

• For example, if vector \( \mathbf{a} = 2 \mathbf{i} + 4 \mathbf{j} \), it's parallel with \( \mathbf{b} = 4 \mathbf{i} + 8 \mathbf{j} \), because \( \mathbf{b} = 2 \times \mathbf{a} \).

In the given problem, none of the vectors were scalar multiples of each other, which means none of the vectors were parallel. This was determined by checking if such a constant could multiply all components from one vector to match another vector.

Dot Product

The dot product is a fundamental operation when working with vectors. It's a way to multiply two vectors to get a scalar, rather than another vector.

The expression for the dot product between two vectors \( \mathbf{a} \text{ and } \mathbf{b} \) is given by:\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\]This formula produces a single number, representing how much one vector extends in the direction of another. It helps us analyze angles between vectors:

• When the dot product is zero, the vectors are perpendicular (or orthogonally related).
• When it isn't zero, they are not perpendicular.

Through the steps outlined in the exercise, we found that the dot products for the vector pairs given were not zero, identifying that none of the vectors were perpendicular.