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What are the direction cosines of a vector in three-dimensional Euclidean space?
What are the direction cosines of a vector in three-dimensional Euclidean space?
- \alpha = \frac{v_x}{v_x^2 - v_y^2 - v_z^2}, \beta = \frac{v_y}{v_x^2 - v_y^2 - v_z^2}, \gamma = \frac{v_z}{v_x^2 - v_y^2 - v_z^2}
- \alpha = \frac{v_x}{v_x^2 \cdot v_y^2 \cdot v_z^2}, \beta = \frac{v_y}{v_x^2 \cdot v_y^2 \cdot v_z^2}, \gamma = \frac{v_z}{v_x^2 \cdot v_y^2 \cdot v_z^2}
- \alpha = \frac{v_x}{v_x^2 + v_y^2 + v_z^2}, \beta = \frac{v_y}{v_x^2 + v_y^2 + v_z^2}, \gamma = \frac{v_z}{v_x^2 + v_y^2 + v_z^2}
- \alpha = \frac{v_x}{\sqrt{v_x^2 + v_y^2 + v_z^2}}, \beta = \frac{v_y}{\sqrt{v_x^2 + v_y^2 + v_z^2}}, \gamma = \frac{v_z}{\sqrt{v_x^2 + v_y^2 + v_z^2}} (correct)
What do the direction cosines represent?
What do the direction cosines represent?
- The angles between the vector and the three negative coordinate axes
- The distances from the origin to the end of the vector
- The magnitudes of the vector components
- The contributions of each component of the basis to a unit vector in that direction (correct)
How are the direction cosines related to the components of the vector?
How are the direction cosines related to the components of the vector?
- \alpha = v_x^2, \beta = v_y^2, \gamma = v_z^2
- \alpha = v_x, \beta = v_y, \gamma = v_z
- \alpha = \frac{v_x}{\sqrt{v_x^2 + v_y^2 + v_z^2}}, \beta = \frac{v_y}{\sqrt{v_x^2 + v_y^2 + v_z^2}}, \gamma = \frac{v_z}{\sqrt{v_x^2 + v_y^2 + v_z^2}} (correct)
- \alpha = \frac{v_x}{v_x^2 + v_y^2 + v_z^2}, \beta = \frac{v_y}{v_x^2 + v_y^2 + v_z^2}, \gamma = \frac{v_z}{v_x^2 + v_y^2 + v_z^2}
If v is a Euclidean vector in three-dimensional space given by $v = 3i - 4j + 5k$, what are the direction cosines?
If v is a Euclidean vector in three-dimensional space given by $v = 3i - 4j + 5k$, what are the direction cosines?
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What are the direction cosines of a unit vector in the direction of the positive x-axis?
What are the direction cosines of a unit vector in the direction of the positive x-axis?
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