What will be the projection of vector A on vector B?

• A = √2(i + j + k)
• B = 2(i + j + k)
• C = √2(i + j)
• D = (i + j)

To find the projection of vector A onto vector B, we can use the formula:

Projection Formula

The projection of vector A onto vector B is given by:

proj_BA = (A · B / |B|²) * B

Step 1: Calculate the Dot Product A · B

First, we need to compute the dot product of A and B.

Given:

• A = √2(i + j + k)
• B = 2(i + j + k)

Calculating the dot product:

A · B = (√2)(2) + (√2)(2) + (√2)(2) = 2√2 + 2√2 + 2√2 = 6√2

Step 2: Calculate |B|²

Next, we find the magnitude squared of vector B:

|B|² = (2)² + (2)² + (2)² = 4 + 4 + 4 = 12

Step 3: Compute the Projection

Now we can substitute these values into the projection formula:

proj_BA = (6√2 / 12) * B = (√2 / 2) * B

Substituting B:

proj_BA = (√2 / 2) * 2(i + j + k) = √2(i + j + k)

Final Result

The projection of vector A onto vector B is:

√2(i + j + k)