7.1 Q-3

Question Statement

Given the vectors:

• $\underline{U} = 2 \underline{\mathbf{i}} - 7 \underline{\mathbf{j}}$
• $\underline{V} = \underline{\mathbf{i}} - 6 \underline{\mathbf{j}}$
• $\underline{W} = -\underline{\mathbf{i}} + \underline{\mathbf{j}}$

Find the following vectors: (i) $\underline{U} + \underline{V} - \underline{W}$
(ii) $2 \underline{U} - 3 \underline{V} + 4 \underline{W}$
(iii) $\frac{1}{2} \underline{U} + \frac{1}{2} \underline{V} + \frac{1}{2} \underline{W}$

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Background and Explanation

In this problem, you are asked to perform vector operations, including addition, subtraction, and scalar multiplication.

• Vector Addition/Subtraction: You simply add or subtract the components of vectors in each direction (i.e., x and y components) to find the resulting vector.
• Scalar Multiplication: When a vector is multiplied by a scalar, you multiply both its x and y components by the scalar.
• These operations follow the rules of algebra, but applied to vectors, which have both magnitude and direction.

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Solution

(i) $\underline{U} + \underline{V} - \underline{W}$

1. Write the vectors in component form:

   • $\underline{U} = 2\hat{i} - 7\hat{j}$
   • $\underline{V} = \hat{i} - 6\hat{j}$
   • $\underline{W} = -\hat{i} + \hat{j}$

2. Perform the vector addition and subtraction:
   $\underline{U} + \underline{V} - \underline{W} = (2\hat{i} - 7\hat{j}) + (\hat{i} - 6\hat{j}) - (-\hat{i} + \hat{j})$

3. Simplify:
   Combine the i-components:
   $2\hat{i} + \hat{i} + \hat{i} = 4\hat{i}$
   Combine the j-components:
   $-7\hat{j} - 6\hat{j} - \hat{j} = -14\hat{j}$

So, the result is:
$\underline{U} + \underline{V} - \underline{W} = 4\hat{i} - 14\hat{j}$

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(ii) $2 \underline{U} - 3 \underline{V} + 4 \underline{W}$

1. Perform scalar multiplication on each vector:

   • $2 \underline{U} = 2(2\hat{i} - 7\hat{j}) = 4\hat{i} - 14\hat{j}$
   • $-3 \underline{V} = -3(\hat{i} - 6\hat{j}) = -3\hat{i} + 18\hat{j}$
   • $4 \underline{W} = 4(-\hat{i} + \hat{j}) = -4\hat{i} + 4\hat{j}$

2. Add the results:
   $(4\hat{i} - 14\hat{j}) + (-3\hat{i} + 18\hat{j}) + (-4\hat{i} + 4\hat{j})$

3. Simplify:
   Combine the i-components:
   $4\hat{i} - 3\hat{i} - 4\hat{i} = -3\hat{i}$
   Combine the j-components:
   $-14\hat{j} + 18\hat{j} + 4\hat{j} = 8\hat{j}$

So, the result is:
$2 \underline{U} - 3 \underline{V} + 4 \underline{W} = -3\hat{i} + 8\hat{j}$

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(iii) $\frac{1}{2} \underline{U} + \frac{1}{2} \underline{V} + \frac{1}{2} \underline{W}$

1. Perform scalar multiplication on each vector:

   • $\frac{1}{2} \underline{U} = \frac{1}{2}(2\hat{i} - 7\hat{j}) = \hat{i} - \frac{7}{2}\hat{j}$
   • $\frac{1}{2} \underline{V} = \frac{1}{2}(\hat{i} - 6\hat{j}) = \frac{1}{2}\hat{i} - 3\hat{j}$
   • $\frac{1}{2} \underline{W} = \frac{1}{2}(-\hat{i} + \hat{j}) = -\frac{1}{2}\hat{i} + \frac{1}{2}\hat{j}$

2. Add the results:
   $\left(\hat{i} - \frac{7}{2}\hat{j}\right) + \left(\frac{1}{2}\hat{i} - 3\hat{j}\right) + \left(-\frac{1}{2}\hat{i} + \frac{1}{2}\hat{j}\right)$

3. Simplify:
   Combine the i-components:
   $\hat{i} + \frac{1}{2}\hat{i} - \frac{1}{2}\hat{i} = \hat{i}$
   Combine the j-components:
   $-\frac{7}{2}\hat{j} - 3\hat{j} + \frac{1}{2}\hat{j} = -6\hat{j}$

So, the result is:
$\frac{1}{2} \underline{U} + \frac{1}{2} \underline{V} + \frac{1}{2} \underline{W} = \hat{i} - 6\hat{j}$

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Key Formulas or Methods Used

• Vector Addition/Subtraction:
  $\underline{U} + \underline{V} = (x_1 + x_2)\hat{i} + (y_1 + y_2)\hat{j}$
  $\underline{U} - \underline{W} = (x_1 - x_2)\hat{i} + (y_1 - y_2)\hat{j}$

• Scalar Multiplication:
  $c\underline{U} = c(x\hat{i} + y\hat{j}) = (cx)\hat{i} + (cy)\hat{j}$

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Summary of Steps

1. For vector addition or subtraction, combine the i and j components separately.
2. For scalar multiplication, multiply each component of the vector by the scalar.
3. Perform the operations step-by-step and simplify the components to get the final result.