7.2 Q-10

Question Statement

We say that two vectors $\underline{\mathbf{V}}$ and $\underline{\mathbf{W}}$ in space are parallel if there exists a scalar $c$ such that: $\underline{\mathbf{V}} = c \underline{\mathbf{W}}$

The vectors point in the same direction if $c > 0$, and in the opposite direction if $c < 0$.

We are given the following vectors:

• $\underline{\mathbf{V}} = 2 \underline{\mathbf{i}} - 4 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}}$
• $\underline{\mathbf{V}} = \underline{\mathbf{i}} - 3 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}}$
• $\underline{\mathbf{W}} = a \underline{\mathbf{i}} + a \underline{\mathbf{j}} - 12 \underline{\mathbf{k}}$

Solve the following:

1. (a) Find the two vectors whose length is 2 and are parallel to $\underline{\mathbf{V}}$.
2. (b) Determine the value of $a$ if $\underline{\mathbf{V}}$ and $\underline{\mathbf{W}}$ are parallel.
3. (c) Find the unit vector in the direction of $\underline{\mathbf{V}}$.
4. (d) Find the values of $a$ and $b$ for which the vectors $3 \underline{i} - \underline{j} + 4 \underline{k}$ and $a \underline{i} + b \underline{j} - 2 \underline{k}$ are parallel.

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Background and Explanation

To solve these problems, we will use the following concepts:

1. Magnitude of a Vector: The magnitude (or length) of a vector $\mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k}$ is: $|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$

2. Unit Vector: A unit vector is a vector with a magnitude of 1. To find the unit vector in the direction of $\mathbf{v}$, we divide each component of $\mathbf{v}$ by its magnitude: $\hat{v} = \frac{\mathbf{v}}{|\mathbf{v}|}$

3. Parallel Vectors: Two vectors are parallel if one is a scalar multiple of the other. That is, $\mathbf{V} = c \mathbf{W}$ for some scalar $c$.

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Solution

(a) Find the two vectors whose length is 2 and are parallel to $\underline{\mathbf{V}}$

1. Find the magnitude of $\underline{\mathbf{V}}$: $|\underline{\mathbf{V}}| = \sqrt{(2)^2 + (-4)^2 + (4)^2} = \sqrt{4 + 16 + 16} = \sqrt{36} = 6$

2. Find the unit vector in the direction of $\underline{\mathbf{V}}$: The unit vector $\hat{V}$ is: $\hat{V} = \frac{\underline{\mathbf{V}}}{|\underline{\mathbf{V}}|} = \frac{1}{6}(2 \underline{\mathbf{i}} - 4 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}})$

3. Find the two vectors with magnitude 2: Multiply the unit vector by 2 and $-2$ to get the two vectors: $2 \hat{V} = \frac{2}{6}(2 \underline{\mathbf{i}} - 4 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}}) = \frac{1}{3}(2 \underline{\mathbf{i}} - 4 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}})$ $-2 \hat{V} = \frac{-2}{6}(2 \underline{\mathbf{i}} - 4 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}}) = -\frac{1}{3}(2 \underline{\mathbf{i}} - 4 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}})$

Thus, the two vectors are: $\frac{1}{3}(2 \underline{\mathbf{i}} - 4 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}}) \quad \text{and} \quad -\frac{1}{3}(2 \underline{\mathbf{i}} - 4 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}})$

(b) Find the value of $a$ if $\underline{\mathbf{V}}$ and $\underline{\mathbf{W}}$ are parallel

For the vectors $\underline{\mathbf{V}}$ and $\underline{\mathbf{W}}$ to be parallel, there must be a scalar $c$ such that: $\underline{\mathbf{V}} = c \underline{\mathbf{W}}$

We are given: $\underline{\mathbf{V}} = \underline{\mathbf{i}} - 3 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}}$ $\underline{\mathbf{W}} = a \underline{\mathbf{i}} + a \underline{\mathbf{j}} - 12 \underline{\mathbf{k}}$

To find $a$, we set up the following ratio of components: $\frac{1}{a} = \frac{-3}{9} = \frac{4}{-12}$

Thus, solving for $a$: $a = -3$

(c) Find the unit vector in the direction of $\underline{\mathbf{V}}$

We already calculated the magnitude of $\underline{\mathbf{V}}$ as $|\underline{\mathbf{V}}| = 6$. Now, to find the unit vector $\hat{V}$, we divide each component of $\underline{\mathbf{V}}$ by its magnitude: $\hat{V} = \frac{\underline{\mathbf{V}}}{|\underline{\mathbf{V}}|} = \frac{1}{6}( \underline{\mathbf{i}} - 3 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}})$

Thus, the unit vector in the direction of $\underline{\mathbf{V}}$ is: $\hat{V} = \frac{1}{6}( \underline{\mathbf{i}} - 3 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}})$

(d) Find the values of $a$ and $b$ for which the vectors $3 \underline{i} - \underline{j} + 4 \underline{k}$ and $a \underline{i} + b \underline{j} - 2 \underline{k}$ are parallel

For the vectors $3 \underline{i} - \underline{j} + 4 \underline{k}$ and $a \underline{i} + b \underline{j} - 2 \underline{k}$ to be parallel, there must exist a scalar $k$ such that: $3 \underline{i} - \underline{j} + 4 \underline{k} = k(a \underline{i} + b \underline{j} - 2 \underline{k})$

Comparing the components: $\frac{3}{a} = \frac{-1}{b} = \frac{4}{-2}$

Solving the ratios:

• $\frac{3}{a} = -2$ implies $a = -\frac{3}{2}$.
• $\frac{-1}{b} = -2$ implies $b = \frac{1}{2}$.

Thus, the values of $a$ and $b$ are: $a = -\frac{3}{2}, \quad b = \frac{1}{2}$

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Key Formulas or Methods Used

• Magnitude of a Vector: $|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$

• Unit Vector: $\hat{v} = \frac{\mathbf{v}}{|\mathbf{v}|}$

• Parallel Vectors: Two vectors are parallel if one is a scalar multiple of the other: $\mathbf{V} = c \mathbf{W}$

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Summary of Steps

1. (a) Find the magnitude of $\mathbf{V}$, then calculate the unit vector. Multiply by 2 and $-2$ to get two vectors parallel to $\mathbf{V}$ with length 2.
2. (b) Set up the parallelism equation and solve for $a$.
3. (c) Find the unit vector of $\mathbf{V}$ by dividing each component by the magnitude.
4. (d) Set up the parallelism equation for the two vectors, solve the ratios for $a$ and $b$, and find their values.