Find modulus,Direction cosines,Unit vectors of (1)4a-2b+c,(2)a-7b+5c,(3)4a-3b+c.
To find the modulus, direction cosines, and unit vector of a given vector, we follow these steps:
Step 1: Find the modulus (magnitude) of the vector.
The modulus of a vector V with components (V₁, V₂, V₃) is given by the formula:
|V| = √(V₁² + V₂² + V₃²)
For example, let's calculate the modulus of the vector V = 4a - 2b + c:
|V| = √((4a)² + (-2b)² + c²)
= √(16a² + 4b² + c²)
Step 2: Find the direction cosines of the vector.
The direction cosines of a vector V with components (V₁, V₂, V₃) are given by:
cos α = V₁ / |V|
cos β = V₂ / |V|
cos γ = V₃ / |V|
For example, let's calculate the direction cosines of the vector V = 4a - 2b + c:
cos α = (4a) / |V|
cos β = (-2b) / |V|
cos γ = c / |V|
Step 3: Find the unit vector of the vector.
The unit vector (also known as the normalized vector) U of a vector V with components (V₁, V₂, V₃) is given by:
U = (V₁ / |V|)â + (V₂ / |V|)b̂ + (V₃ / |V|)ĉ
where â, b̂, and ĉ are the unit vectors along the respective axes.
For example, let's calculate the unit vector of the vector V = 4a - 2b + c:
U = ((4a) / |V|)â + ((-2b) / |V|)b̂ + (c / |V|)ĉ
Now, you can apply these steps to find the modulus, direction cosines, and unit vectors for the vectors (1) 4a - 2b + c, (2) a - 7b + 5c, and (3) 4a - 3b + c.