(6) Consider the vectors Vector a=(1,2,3) and Vector b=(-3,0,2).
(a) Determine the projection of Vector a onto Vector b.
(b) A unit vector is a vector of length 1. Determine a simplified unit vector in the same direction as Vector a .
projection of a onto b = a dot b /|b|
= ( (1,2,3) dot (-3,0,2) )/√(9+0+4)
= (-3 + 0 + 6)/√13 = 3/√13
unit vector a = (1/√(1+4+9)(1,2,3)
= (1/√14 , 2/√14 , 3/√14)
(a) To determine the projection of Vector a onto Vector b, we can use the formula:
proj_b(a) = (a · b / ||b||^2) * b
where · represents the dot product and ||b|| represents the magnitude of Vector b.
First, let's calculate the dot product of Vector a and Vector b:
a · b = (1 * -3) + (2 * 0) + (3 * 2)
= -3 + 0 + 6
= 3
Next, we need to find the magnitude of Vector b:
||b|| = √((-3)^2 + 0^2 + 2^2)
= √(9 + 0 + 4)
= √13
Now, we can substitute the values into the projection formula:
proj_b(a) = (3 / (√13)^2) * (-3, 0, 2)
= (3 / 13) * (-3, 0, 2)
= (-9/13, 0, 6/13)
Therefore, the projection of Vector a onto Vector b is (-9/13, 0, 6/13).
(b) To determine a simplified unit vector in the same direction as Vector a, we can use the formula:
unit_a = a / ||a||
where ||a|| represents the magnitude of Vector a.
Let's calculate the magnitude of Vector a:
||a|| = √(1^2 + 2^2 + 3^2)
= √(1 + 4 + 9)
= √14
Now, we can substitute the values into the unit vector formula:
unit_a = (1/√14, 2/√14, 3/√14)
Therefore, a simplified unit vector in the same direction as Vector a is (1/√14, 2/√14, 3/√14).
To determine the projection of Vector a onto Vector b, we can use the formula:
proj"a" "b" = ( "a" · "b" ) / | "b" |
Where "a" · "b" is the dot product of Vector a and Vector b, and | "b" | is the magnitude (length) of Vector b.
Step 1: Calculate the dot product of Vector a and Vector b.
The dot product is calculated by multiplying the corresponding components of the two vectors and summing the products.
"a" · "b" = (1 * (-3)) + (2 * 0) + (3 * 2) = -3 + 0 + 6 = 3
Step 2: Calculate the magnitude (length) of Vector b.
The magnitude is determined by taking the square root of the sum of the squares of its components.
| "b" | = sqrt( (-3)^2 + 0^2 + 2^2 ) = sqrt(9 + 0 + 4) = sqrt(13)
Step 3: Calculate the projection of Vector a onto Vector b.
proj"a" "b" = ( "a" · "b" ) / | "b" | = 3 / sqrt(13)
Therefore, the projection of Vector a onto Vector b is 3 / sqrt(13).
To determine a simplified unit vector in the same direction as Vector a, we can divide each component of Vector a by its magnitude.
Step 1: Calculate the magnitude (length) of Vector a.
| "a" | = sqrt( 1^2 + 2^2 + 3^2 ) = sqrt(1 + 4 + 9) = sqrt(14)
Step 2: Divide each component of Vector a by its magnitude.
Unit vector = (1 / sqrt(14), 2 / sqrt(14), 3 / sqrt(14))
Therefore, a simplified unit vector in the same direction as Vector a is (1 / sqrt(14), 2 / sqrt(14), 3 / sqrt(14)).