7.3 Q-5

Question Statement

Given the conditions:

$$\underline{\mathbf{V}} \cdot \underline{\mathbf{i}} = 0, \quad \underline{\mathbf{V}} \cdot \underline{\mathbf{j}} = 0, \quad \underline{\mathbf{V}} \cdot \underline{\mathbf{k}} = 0$$

Find the vector $\underline{\mathbf{V}}$, where $\underline{\mathbf{V}} = a \underline{\mathbf{i}} + b \underline{\mathbf{j}} + c \underline{\mathbf{k}}$.

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Background and Explanation

In this problem, we are given the conditions that the vector $\underline{\mathbf{V}}$ is orthogonal (perpendicular) to the unit vectors $\underline{\mathbf{i}}$, $\underline{\mathbf{j}}$, and $\underline{\mathbf{k}}$. The dot product between two perpendicular vectors is zero.

• The dot product formula for two vectors $\mathbf{U} = (u_1, u_2, u_3)$ and $\mathbf{V} = (v_1, v_2, v_3)$ is:

$$\mathbf{U} \cdot \mathbf{V} = u_1 v_1 + u_2 v_2 + u_3 v_3$$

• The conditions $\underline{\mathbf{V}} \cdot \underline{\mathbf{i}} = 0$, $\underline{\mathbf{V}} \cdot \underline{\mathbf{j}} = 0$, and $\underline{\mathbf{V}} \cdot \underline{\mathbf{k}} = 0$ imply that the vector $\underline{\mathbf{V}}$ must be the null vector, as it has no component in the directions of $\underline{\mathbf{i}}$, $\underline{\mathbf{j}}$, or $\underline{\mathbf{k}}$.

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Solution

We are given the vector $\underline{\mathbf{V}} = a \underline{\mathbf{i}} + b \underline{\mathbf{j}} + c \underline{\mathbf{k}}$, and we need to solve for $a$, $b$, and $c$ under the conditions that:

1. Dot product of $\underline{\mathbf{V}}$ and $\underline{\mathbf{i}}$:

$$\underline{\mathbf{V}} \cdot \underline{\mathbf{i}} = 0$$

Substitute the components of $\underline{\mathbf{V}}$ and $\underline{\mathbf{i}}$:

$$(a \underline{\mathbf{i}} + b \underline{\mathbf{j}} + c \underline{\mathbf{k}}) \cdot (\underline{\mathbf{i}} + 0 \underline{\mathbf{j}} + 0 \underline{\mathbf{k}}) = 0$$

Simplifying:

$$a(1) + b(0) + c(0) = 0$$ $$a = 0$$

2. Dot product of $\underline{\mathbf{V}}$ and $\underline{\mathbf{j}}$:

$$\underline{\mathbf{V}} \cdot \underline{\mathbf{j}} = 0$$

Substitute the components of $\underline{\mathbf{V}}$ and $\underline{\mathbf{j}}$:

$$a(0) + b(1) + c(0) = 0$$ $$b = 0$$

3. Dot product of $\underline{\mathbf{V}}$ and $\underline{\mathbf{k}}$:

$$\underline{\mathbf{V}} \cdot \underline{\mathbf{k}} = 0$$

Substitute the components of $\underline{\mathbf{V}}$ and $\underline{\mathbf{k}}$:

$$a(0) + b(0) + c(1) = 0$$ $$c = 0$$

4. Final result:

From the above calculations, we find that $a = 0$, $b = 0$, and $c = 0$. Thus, the vector $\underline{\mathbf{V}}$ is:

$$\underline{\mathbf{V}} = 0 \underline{\mathbf{i}} + 0 \underline{\mathbf{j}} + 0 \underline{\mathbf{k}} = \mathbf{0}$$

Therefore, $\underline{\mathbf{V}}$ is the null vector.

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Key Formulas or Methods Used

• Dot product of two vectors $\mathbf{U} = (u_1, u_2, u_3)$ and $\mathbf{V} = (v_1, v_2, v_3)$:

$$\mathbf{U} \cdot \mathbf{V} = u_1 v_1 + u_2 v_2 + u_3 v_3$$

• Condition for perpendicular vectors: Two vectors are perpendicular if their dot product is zero:

$$\mathbf{U} \cdot \mathbf{V} = 0$$

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Summary of Steps

1. Write down the given vector $\underline{\mathbf{V}} = a \underline{\mathbf{i}} + b \underline{\mathbf{j}} + c \underline{\mathbf{k}}$.
2. Set up the dot product equations for each of the unit vectors $\underline{\mathbf{i}}$, $\underline{\mathbf{j}}$, and $\underline{\mathbf{k}}$.
3. Solve each equation to find $a$, $b$, and $c$.
4. Substitute the values $a = 0$, $b = 0$, and $c = 0$ into the vector $\underline{\mathbf{V}}$.
5. Conclude that $\underline{\mathbf{V}}$ is the null vector $\mathbf{0}$.