Vector Algebra
Direction Cosines and Direction Ratios
Grade 12
Question:
<p>The direction cosines of the vector 3<strong>i</strong> - 4<strong>j</strong> + 5<strong>k</strong> are</p>
<p>(a) \(\frac{3}{5\sqrt{2}}, -\frac{4}{5\sqrt{2}}, \frac{1}{\sqrt{2}}\)</p>
<p>(b) \(\frac{3}{5\sqrt{2}}, \frac{-4}{5\sqrt{2}}, \frac{1}{\sqrt{2}}\)</p>
<p>(c) \(\frac{3}{2\sqrt{2}}, \frac{-4}{2\sqrt{2}}, \frac{1}{2\sqrt{2}}\)</p>
<p>(d) \(\frac{3}{5\sqrt{2}}, \frac{4}{5\sqrt{2}}, \frac{1}{5\sqrt{2}}\)</p>
Step-by-Step Solution
Key Concept: Direction cosines of a vector are obtained by dividing each component by the magnitude of the vector.
Solution: The magnitude of vector = \(\sqrt{3^2 + (-4)^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} = 5\sqrt{2}\) Direction cosines are: \(\left(\frac{3}{5\sqrt{2}}, \frac{-4}{5\sqrt{2}}, \frac{5}{5\sqrt{2}}\right) = \left(\frac{3}{5\sqrt{2}}, -\frac{4}{5\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\)
Correct Answer: A