find the projection of vector A=I-2j+k in the vector B=4i-4j+7k
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To find the projection of vector A onto vector B, we can use the following formula:
Projection of A onto B = (A • B) / ||B|| * B
Where:
- A • B represents the dot product of vectors A and B,
- ||B|| represents the magnitude (length) of vector B, and
- * represents scalar multiplication between the dot product result and vector B.
Let's calculate the projection of A onto B step by step:
Step 1: Calculate the dot product of vectors A and B:
A • B = (I-2j+k) • (4i-4j+7k)
To find the dot product, we multiply corresponding components and sum them:
A • B = (1 * 4) + (-2 * -4) + (1 * 7)
= 4 + 8 + 7
= 19
Therefore, A • B = 19.
Step 2: Calculate the magnitude of vector B:
||B|| = √(4^2 + (-4)^2 + 7^2)
= √(16 + 16 + 49)
= √(81)
= 9
Therefore, ||B|| = 9.
Step 3: Calculate the projection of A onto B:
Projection of A onto B = (A • B) / ||B|| * B
Projection of A onto B = (19 / 9) * (4i-4j+7k)
= (19/9) * 4i + (19/9) * -4j + (19/9) * 7k
Simplifying, we get:
Projection of A onto B = 76/9 i - 76/9 j + 133/9 k
Therefore, the projection of vector A onto vector B is (76/9)i - (76/9)j + (133/9)k.
To find the projection of vector A onto vector B, you can use the following formula:
proj(A onto B) = ((A · B) / ||B||^2) * B
where · denotes the dot product and ||B|| represents the magnitude of vector B.
First, let's calculate the dot product of vectors A and B:
A · B = (I-2j+k) · (4i-4j+7k)
= (1*4) + (-2*-4) + (1*7)
= 4 + 8 + 7
= 19
Next, let's calculate the magnitude of vector B:
||B|| = √(4^2 + (-4)^2 + 7^2)
= √(16 + 16 + 49)
= √81
= 9
Now we can substitute these values into the projection formula:
proj(A onto B) = ((19) / (9^2)) * (4i-4j+7k)
= (19 / 81) * (4i-4j+7k)
Therefore, the projection of vector A onto vector B is (19 / 81) * (4i-4j+7k).