To find the magnitude and direction of the vectors \(A\), \(C\), and \(D\), we will follow these steps:
Given Vectors:
- \( A = 2\mathbf{i} - 5\mathbf{j} + 3\mathbf{k} , \text{m} \)
- \( B = 6\mathbf{i} + 9\mathbf{j} - 13\mathbf{k} , \text{m} \)
Step 1: Calculate the resultant vectors
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Vector C: \( C = A + 4B \) \[ C = (2\mathbf{i} - 5\mathbf{j} + 3\mathbf{k}) + 4(6\mathbf{i} + 9\mathbf{j} - 13\mathbf{k}) \] \[ = (2 + 24)\mathbf{i} + (-5 + 36)\mathbf{j} + (3 - 52)\mathbf{k} \] \[ = 26\mathbf{i} + 31\mathbf{j} - 49\mathbf{k} , \text{m} \]
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Vector D: \( D = 2A - 3B \) \[ D = 2(2\mathbf{i} - 5\mathbf{j} + 3\mathbf{k}) - 3(6\mathbf{i} + 9\mathbf{j} - 13\mathbf{k}) \] \[ = (4 - 18)\mathbf{i} + (-10 - 27)\mathbf{j} + (6 + 39)\mathbf{k} \] \[ = -14\mathbf{i} - 37\mathbf{j} + 45\mathbf{k} , \text{m} \]
Step 2: Calculate the magnitude and direction of A, C, and D
Magnitude of a vector \( \mathbf{V} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \) is calculated as: \[ |\mathbf{V}| = \sqrt{a^2 + b^2 + c^2} \]
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Magnitude of A: \[ |A| = \sqrt{2^2 + (-5)^2 + 3^2} = \sqrt{4 + 25 + 9} = \sqrt{38} \approx 6.16 , \text{m} \]
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Magnitude of C: \[ |C| = \sqrt{26^2 + 31^2 + (-49)^2} = \sqrt{676 + 961 + 2401} = \sqrt{4038} \approx 63.6 , \text{m} \]
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Magnitude of D: \[ |D| = \sqrt{(-14)^2 + (-37)^2 + 45^2} = \sqrt{196 + 1369 + 2025} = \sqrt{3590} \approx 59.9 , \text{m} \]
Direction:
The direction can be represented as a unit vector for each vector. The unit vector \( \hat{V} \) of a vector \( \mathbf{V} \) is given by: \[ \hat{V} = \frac{\mathbf{V}}{|\mathbf{V}|} \]
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Direction of A: \[ \hat{A} = \frac{A}{|A|} = \frac{2\mathbf{i} - 5\mathbf{j} + 3\mathbf{k}}{\sqrt{38}} \approx \left( \frac{2}{\sqrt{38}}, \frac{-5}{\sqrt{38}}, \frac{3}{\sqrt{38}} \right) \]
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Direction of C: \[ \hat{C} = \frac{C}{|C|} = \frac{26\mathbf{i} + 31\mathbf{j} - 49\mathbf{k}}{\sqrt{4038}} \approx \left( \frac{26}{\sqrt{4038}}, \frac{31}{\sqrt{4038}}, \frac{-49}{\sqrt{4038}} \right) \]
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Direction of D: \[ \hat{D} = \frac{D}{|D|} = \frac{-14\mathbf{i} - 37\mathbf{j} + 45\mathbf{k}}{\sqrt{3590}} \approx \left( \frac{-14}{\sqrt{3590}}, \frac{-37}{\sqrt{3590}}, \frac{45}{\sqrt{3590}} \right) \]
Summary:
- Vector A: Magnitude \(\approx 6.16 , \text{m}\), Direction \(\left( \frac{2}{\sqrt{38}}, \frac{-5}{\sqrt{38}}, \frac{3}{\sqrt{38}} \right)\)
- Vector C: Magnitude \(\approx 63.6 , \text{m}\), Direction \(\left( \frac{26}{\sqrt{4038}}, \frac{31}{\sqrt{4038}}, \frac{-49}{\sqrt{4038}} \right)\)
- Vector D: Magnitude \(\approx 59.9 , \text{m}\), Direction \(\left( \frac{-14}{\sqrt{3590}}, \frac{-37}{\sqrt{3590}}, \frac{45}{\sqrt{3590}} \right)\)