let U = ( 1,-2, 3) find the direction cosines of U
To find the direction cosines of U, we need to find the cosines of the angles that U makes with the x-axis, y-axis, and z-axis.
The direction cosines can be found using the following formulas:
l = Ux / ||U||
m = Uy / ||U||
n = Uz / ||U||
Where:
Ux, Uy, and Uz are the components of U in the x, y, and z directions respectively.
||U|| is the magnitude of U.
Given that U = (1, -2, 3), we can find ||U|| as:
||U|| = sqrt(1^2 + (-2)^2 + 3^2)
= sqrt(1 + 4 + 9)
= sqrt(14)
Now, we can find the direction cosines:
l = 1 / sqrt(14)
m = -2 / sqrt(14)
n = 3 / sqrt(14)
Therefore, the direction cosines of U are:
l ≈ 0.267
m ≈ -0.534
n ≈ 0.801
To find the direction cosines of a vector, we can use the following formula:
Direction cosine (cosθ) = uᵢ / ||U||
Where:
- uᵢ is the i-th component of vector U
- ||U|| is the magnitude (or length) of vector U
For vector U = (1, -2, 3), let's find its magnitude first:
||U|| = sqrt(1² + (-2)² + 3²)
= sqrt(1 + 4 + 9)
= sqrt(14)
Now, let's find the direction cosines for each component:
Direction cosine (cosθ₁) = 1 / sqrt(14)
Direction cosine (cosθ₂) = -2 / sqrt(14)
Direction cosine (cosθ₃) = 3 / sqrt(14)
Hence, the direction cosines of U are:
cosθ₁ = 1 / sqrt(14)
cosθ₂ = -2 / sqrt(14)
cosθ₃ = 3 / sqrt(14)