7.2 Q-8

Question Statement

Given the vectors:

• $\underline{U} = 2 \underline{\mathbf{i}} + 3 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}}$
• $\underline{\mathbf{V}} = -\underline{\mathbf{i}} + 3 \underline{\mathbf{j}} - \underline{\mathbf{k}}$
• $\underline{\mathbf{W}} = \underline{\mathbf{i}} + 6 \underline{\mathbf{j}} + z \underline{\mathbf{k}}$

These vectors represent the sides of a triangle. Find the value of $z$ such that the sum of these vectors equals zero.

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Background and Explanation

In this problem, we are given three vectors that represent the sides of a triangle. A key property of vectors in a triangle is that the sum of the vectors representing the sides must equal zero. This is derived from the triangle law of vector addition, which states that if three vectors represent the sides of a triangle, then: $\mathbf{U} + \mathbf{V} + \mathbf{W} = 0$

To solve for $z$, we will use vector addition and solve for the unknown component in the $z$-direction.

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Solution

We are given the equation: $\mathbf{U} + \mathbf{V} + \mathbf{W} = 0$

Substitute the components of the vectors $\mathbf{U}$, $\mathbf{V}$, and $\mathbf{W}$: $(2 \underline{\mathbf{i}} + 3 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}}) + (-\underline{\mathbf{i}} + 3 \underline{\mathbf{j}} - \underline{\mathbf{k}}) + (\underline{\mathbf{i}} + 6 \underline{\mathbf{j}} + z \underline{\mathbf{k}}) = 0$

Now, combine the components of the vectors:

• For $\mathbf{i}$: $2 - 1 + 1 = 2 \underline{\mathbf{i}}$
• For $\mathbf{j}$: $3 + 3 + 6 = 12 \underline{\mathbf{j}}$
• For $\mathbf{k}$: $4 - 1 + z = (3 + z) \underline{\mathbf{k}}$

So the equation becomes: $2 \underline{\mathbf{i}} + 12 \underline{\mathbf{j}} + (3 + z) \underline{\mathbf{k}} = 0$

Now, compare the components on both sides of the equation. Since the vectors on the right-hand side of the equation are zero, we compare the coefficients of $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$:

1. For $\mathbf{i}$: $2 = 0$ (this is automatically satisfied since there is no $\mathbf{i}$ term on the right side).

2. For $\mathbf{j}$: $12 = 0$ (this is satisfied as there is no $\mathbf{j}$ term on the right side).

3. For $\mathbf{k}$: $(3 + z) = 0$

Solving for $z$: $3 + z = 0$

Thus: $z = -3$

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Key Formulas or Methods Used

• Triangle Law of Vector Addition: $\mathbf{U} + \mathbf{V} + \mathbf{W} = 0$

• Vector Addition: Adding the corresponding components of vectors to solve for the unknowns.

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Summary of Steps

1. Write the equation for the sum of the vectors: $\mathbf{U} + \mathbf{V} + \mathbf{W} = 0$.
2. Substitute the components of each vector into the equation.
3. Combine like terms (components of $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$).
4. Solve for the unknown $z$ by setting the $\mathbf{k}$ component equal to zero.
5. Find that $z = -3$.