Find the modulus direction cosin unit vector 1) 4a-2b+3c (2) a-7b+5c (3)4a-3b+c
To find the modulus of a vector, we need to find the magnitude of the vector. In this case, we need to find the magnitude of each given vector.
To calculate the magnitude of a vector, we use the formula:
|v| = √(v₁² + v₂² + v₃²)
Let's calculate the magnitude of each given vector:
1) For the vector 4a - 2b + 3c:
|4a - 2b + 3c| = √((4a)² + (-2b)² + (3c)²) = √(16a² + 4b² + 9c²)
2) For the vector a - 7b + 5c:
|a - 7b + 5c| = √((a)² + (-7b)² + (5c)²) = √(a² + 49b² + 25c²)
3) For the vector 4a - 3b + c:
|4a - 3b + c| = √((4a)² + (-3b)² + (c)²) = √(16a² + 9b² + c²)
Next, to find the direction cosines of a vector, we divide each component of the vector by its magnitude. The direction cosines represent the direction of the vector in terms of ratios of its components.
The direction cosines of a vector v = (v₁, v₂, v₃) are given by:
l₁ = v₁ / |v|
l₂ = v₂ / |v|
l₃ = v₃ / |v|
Let's calculate the direction cosines of each given vector:
1) For the vector 4a - 2b + 3c:
l₁ = 4a / √(16a² + 4b² + 9c²)
l₂ = -2b / √(16a² + 4b² + 9c²)
l₃ = 3c / √(16a² + 4b² + 9c²)
2) For the vector a - 7b + 5c:
l₁ = a / √(a² + 49b² + 25c²)
l₂ = -7b / √(a² + 49b² + 25c²)
l₃ = 5c / √(a² + 49b² + 25c²)
3) For the vector 4a - 3b + c:
l₁ = 4a / √(16a² + 9b² + c²)
l₂ = -3b / √(16a² + 9b² + c²)
l₃ = c / √(16a² + 9b² + c²)
And finally, the unit vector is obtained by dividing each component of the vector by its magnitude. The unit vector has a magnitude equal to 1.
The unit vector u₁ = (l₁, l₂, l₃) is given by:
u₁ = (l₁ / |v|, l₂ / |v|, l₃ / |v|)
Let's calculate the unit vector for each given vector:
1) For the vector 4a - 2b + 3c:
u₁ = (4a / √(16a² + 4b² + 9c²), -2b / √(16a² + 4b² + 9c²), 3c / √(16a² + 4b² + 9c²))
2) For the vector a - 7b + 5c:
u₁ = (a / √(a² + 49b² + 25c²), -7b / √(a² + 49b² + 25c²), 5c / √(a² + 49b² + 25c²))
3) For the vector 4a - 3b + c:
u₁ = (4a / √(16a² + 9b² + c²), -3b / √(16a² + 9b² + c²), c / √(16a² + 9b² + c²))
Thus, we have calculated the modulus, direction cosines, and unit vector for each given vector.