7.4 Q-5

Question Statement

Determine which vectors, if any, are perpendicular or parallel in the following cases:

(i)

$$\underline{\mathbf{U}} = 5 \underline{\mathbf{i}} - \underline{\mathbf{j}} + \underline{\mathbf{k}}, \quad \underline{\mathbf{V}} = \underline{\mathbf{j}} - 5 \underline{\mathbf{k}}, \quad \underline{\mathbf{W}} = -15 \underline{\mathbf{i}} + 3 \underline{\mathbf{j}} - 3 \underline{\mathbf{k}}$$

(ii)

$$\underline{\mathbf{U}} = \underline{\mathbf{i}} + 2 \underline{\mathbf{j}} - \underline{\mathbf{k}}, \quad \underline{\mathbf{V}} = - \underline{\mathbf{i}} + \underline{\mathbf{j}} + \underline{\mathbf{k}}, \quad \underline{\mathbf{W}} = -\frac{\pi}{2} \underline{\mathbf{i}} - \pi \underline{\mathbf{j}} + \frac{\pi}{2} \underline{\mathbf{k}}$$

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Background and Explanation

In vector analysis, two vectors are considered perpendicular if their dot product is zero, i.e.,

$$\underline{\mathbf{A}} \cdot \underline{\mathbf{B}} = 0$$

Two vectors are parallel if one is a scalar multiple of the other. To determine if two vectors are perpendicular or parallel, we can use these conditions and calculate the dot products as shown below.

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Solution

(i) First set of vectors:

Step 1: Check if $\underline{\mathbf{U}}$ and $\underline{\mathbf{V}}$ are perpendicular

We calculate the dot product between $\underline{\mathbf{U}}$ and $\underline{\mathbf{V}}$:

$$\underline{\mathbf{U}} \cdot \underline{\mathbf{V}} = (5 \underline{\mathbf{i}} - \underline{\mathbf{j}} + \underline{\mathbf{k}}) \cdot (\underline{\mathbf{j}} - 5 \underline{\mathbf{k}})$$

Expanding the terms:

$$= 5 \cdot 0 + (-1) \cdot 1 + 1 \cdot (-5) = 0 - 1 - 5 = -6$$

Since the dot product is not zero ($-6 \neq 0$), $\underline{\mathbf{U}}$ and $\underline{\mathbf{V}}$ are not perpendicular.

Step 2: Check if $\underline{\mathbf{U}}$ and $\underline{\mathbf{W}}$ are perpendicular

Now, calculate the dot product between $\underline{\mathbf{U}}$ and $\underline{\mathbf{W}}$:

$$\underline{\mathbf{U}} \cdot \underline{\mathbf{W}} = (5 \underline{\mathbf{i}} - \underline{\mathbf{j}} + \underline{\mathbf{k}}) \cdot (-15 \underline{\mathbf{i}} + 3 \underline{\mathbf{j}} - 3 \underline{\mathbf{k}})$$

Expanding the terms:

$$= 5 \cdot (-15) + (-1) \cdot 3 + 1 \cdot (-3) = -75 - 3 - 3 = -81$$

Since the dot product is not zero ($-81 \neq 0$), $\underline{\mathbf{U}}$ and $\underline{\mathbf{W}}$ are not perpendicular.

Step 3: Check if $\underline{\mathbf{V}}$ and $\underline{\mathbf{W}}$ are perpendicular

Next, we calculate the dot product between $\underline{\mathbf{V}}$ and $\underline{\mathbf{W}}$:

$$\underline{\mathbf{V}} \cdot \underline{\mathbf{W}} = (\underline{\mathbf{j}} - 5 \underline{\mathbf{k}}) \cdot (-15 \underline{\mathbf{i}} + 3 \underline{\mathbf{j}} - 3 \underline{\mathbf{k}})$$

Expanding the terms:

$$= 0 \cdot (-15) + 1 \cdot 3 + (-5) \cdot (-3) = 0 + 3 + 15 = 18$$

Since the dot product is not zero ($18 \neq 0$), $\underline{\mathbf{V}}$ and $\underline{\mathbf{W}}$ are not perpendicular.

Step 4: Check if $\underline{\mathbf{W}}$ is parallel to $\underline{\mathbf{U}}$

We notice that:

$$\underline{\mathbf{W}} = -15 \underline{\mathbf{i}} + 3 \underline{\mathbf{j}} - 3 \underline{\mathbf{k}} = -3(5 \underline{\mathbf{i}} - \underline{\mathbf{j}} + \underline{\mathbf{k}}) = -3 \underline{\mathbf{U}}$$

Thus, $\underline{\mathbf{W}} | \underline{\mathbf{U}}$, meaning $\underline{\mathbf{W}}$ and $\underline{\mathbf{U}}$ are parallel.

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(ii) Second set of vectors:

Step 1: Check if $\underline{\mathbf{U}}$ and $\underline{\mathbf{V}}$ are perpendicular

We calculate the dot product between $\underline{\mathbf{U}}$ and $\underline{\mathbf{V}}$:

$$\underline{\mathbf{U}} \cdot \underline{\mathbf{V}} = (\underline{\mathbf{i}} + 2 \underline{\mathbf{j}} - \underline{\mathbf{k}}) \cdot (-\underline{\mathbf{i}} + \underline{\mathbf{j}} + \underline{\mathbf{k}})$$

Expanding the terms:

$$= 1 \cdot (-1) + 2 \cdot 1 + (-1) \cdot 1 = -1 + 2 - 1 = 0$$

Since the dot product is zero, $\underline{\mathbf{U}}$ and $\underline{\mathbf{V}}$ are perpendicular.

Step 2: Check if $\underline{\mathbf{U}}$ and $\underline{\mathbf{W}}$ are perpendicular

Next, we calculate the dot product between $\underline{\mathbf{U}}$ and $\underline{\mathbf{W}}$:

$$\underline{\mathbf{U}} \cdot \underline{\mathbf{W}} = (\underline{\mathbf{i}} + 2 \underline{\mathbf{j}} - \underline{\mathbf{k}}) \cdot \left(-\frac{\pi}{2} \underline{\mathbf{i}} - \pi \underline{\mathbf{j}} + \frac{\pi}{2} \underline{\mathbf{k}}\right)$$

Expanding the terms:

$$= 1 \cdot \left(-\frac{\pi}{2}\right) + 2 \cdot (-\pi) + (-1) \cdot \frac{\pi}{2} = -\frac{\pi}{2} - 2\pi - \frac{\pi}{2} = -3\pi$$

Since the dot product is not zero ($-3\pi \neq 0$), $\underline{\mathbf{U}}$ and $\underline{\mathbf{W}}$ are not perpendicular.

Step 3: Check if $\underline{\mathbf{V}}$ and $\underline{\mathbf{W}}$ are perpendicular

We calculate the dot product between $\underline{\mathbf{V}}$ and $\underline{\mathbf{W}}$:

$$\underline{\mathbf{V}} \cdot \underline{\mathbf{W}} = (-\underline{\mathbf{i}} + \underline{\mathbf{j}} + \underline{\mathbf{k}}) \cdot \left(-\frac{\pi}{2} \underline{\mathbf{i}} - \pi \underline{\mathbf{j}} + \frac{\pi}{2} \underline{\mathbf{k}}\right)$$

Expanding the terms:

$$= -1 \cdot \left(-\frac{\pi}{2}\right) + 1 \cdot (-\pi) + 1 \cdot \frac{\pi}{2} = \frac{\pi}{2} - \pi + \frac{\pi}{2} = 0$$

Since the dot product is zero, $\underline{\mathbf{V}}$ and $\underline{\mathbf{W}}$ are perpendicular.

Step 4: Check if $\underline{\mathbf{W}}$ is parallel to $\underline{\mathbf{U}}$

We observe that:

$$\underline{\mathbf{W}} = -\frac{\pi}{2} \underline{\mathbf{i}} - \pi \underline{\mathbf{j}} + \frac{\pi}{2} \underline{\mathbf{k}} = \frac{\pi}{2} (\underline{\mathbf{i}} + 2 \underline{\mathbf{j}} - \underline{\mathbf{k}}) = \frac{\pi}{2} \underline{\mathbf{U}}$$

Thus, $\underline{\mathbf{W}} | \underline{\mathbf{U}}$, meaning $\underline{\mathbf{W}}$ and $\underline{\mathbf{U}}$ are parallel.

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Key Formulas or Methods Used

• Dot Product: To determine perpendicularity, use the formula:

$$\underline{\mathbf{A}} \cdot \underline{\mathbf{B}} = A_x B_x + A_y B_y + A_z B_z$$

If $\underline{\mathbf{A}} \cdot \underline{\mathbf{B}} = 0$, then the vectors are perpendicular.

• Parallel Vectors: Vectors $\underline{\mathbf{A}}$ and $\underline{\mathbf{B}}$ are parallel if:

$$\underline{\mathbf{A}} = k \underline{\mathbf{B}} \quad \text{for some scalar } k$$

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Summary of Steps

1. For case (i):
   • $\underline{\mathbf{U}}$ and $\underline{\mathbf{V}}$ are not perpendicular.
   • $\underline{\mathbf{U}}$ and $\underline{\mathbf{W}}$ are not perpendicular.
   • $\underline{\mathbf{V}}$ and $\underline{\mathbf{W}}$ are not perpendicular.
   • $\underline{\mathbf{W}}$ is parallel to $\underline{\mathbf{U}}$.
2. For case (ii):
   • $\underline{\mathbf{U}}$ and $\underline{\mathbf{V}}$ are perpendicular.
   • $\underline{\mathbf{U}}$ and $\underline{\mathbf{W}}$ are not perpendicular.
   • $\underline{\mathbf{V}}$ and $\underline{\mathbf{W}}$ are perpendicular.
   • $\underline{\mathbf{W}}$ is parallel to $\underline{\mathbf{U}}$.