If Vector a = (5, -5, 3), Vector b = (1, 2, 3), determin proj (Vector a onto Vector b).
the projection of vector a on vector b
= (a dot b)/|b|
= (5-10+9)/√(1+4+9)
= 4/√14
or
= 4√14 /14 = 2√14/7
Oh, projection time! Let me put on my projection hat and juggle some vectors!
To find the projection of vector a onto vector b, we can use the formula:
proj(a onto b) = (a · b / ||b||^2) * b
Where "·" denotes the dot product and "|| ||" denotes the magnitude of a vector.
Now, let's get started. First, we need to calculate the dot product of a and b:
a · b = (5 * 1) + (-5 * 2) + (3 * 3) = 5 - 10 + 9 = 4
Next, we need to find the magnitude squared of vector b:
||b||^2 = (1^2) + (2^2) + (3^2) = 1 + 4 + 9 = 14
Now, we can plug these values into the formula:
proj(a onto b) = (4 / 14) * (1, 2, 3)
Simplifying further, we get:
proj(a onto b) ≈ (0.2857, 0.5714, 0.8571)
So, the projection of vector a onto vector b is approximately (0.2857, 0.5714, 0.8571). Clowns out! 🤡🤹
To determine the projection of Vector a onto Vector b, we can use the formula:
proj a onto b = (a · b) / ||b||^2 * b
Where
- · represents the dot product of the two vectors
- ||b|| represents the magnitude (or length) of Vector b
- b represents Vector b
Let's calculate it step-by-step:
Step 1: Calculate the dot product of Vector a and Vector b
a · b = (5 * 1) + (-5 * 2) + (3 * 3) = 5 - 10 + 9 = 4
Step 2: Calculate the magnitude of Vector b
||b|| = √(1^2 + 2^2 + 3^2) = √(1 + 4 + 9) = √14
Step 3: Calculate the projection of Vector a onto Vector b
proj a onto b = (a · b) / ||b||^2 * b
= 4 / 14 * (1, 2, 3)
= (4/14, 8/14, 12/14)
Therefore, the projection of Vector a onto Vector b is approximately (0.29, 0.57, 0.86).
To find the projection of vector a onto vector b, we can use the formula:
proj(a onto b) = (a · b) / ||b||^2 * b
where · represents the dot product and ||b||^2 represents the squared magnitude of vector b.
Step 1: Calculate the dot product of a and b.
a · b = (5 * 1) + (-5 * 2) + (3 * 3) = 5 - 10 + 9 = 4
Step 2: Calculate the squared magnitude of vector b.
||b||^2 = (1^2) + (2^2) + (3^2) = 1 + 4 + 9 = 14
Step 3: Calculate the projection vector.
proj(a onto b) = (4 / 14) * (1, 2, 3) = (4/14, 8/14, 12/14)
Simplifying the resulting vector:
proj(a onto b) ≈ (0.286, 0.571, 0.857)