How do you find the scalar and vector projections of b onto a where a = (5, 12) and b = (4, 6)?

To find the scalar and vector projections of vector b onto vector a, we start with the vectors given:

a = (5, 12)
b = (4, 6)

Step 1: Calculate the scalar projection of b onto a.

The formula for the scalar projection of vector b onto vector a is given by:

scalar projection = rac{ba}{||a||}

First, we find the dot product ba:

ba = (4)(5) + (6)(12) = 20 + 72 = 92

Next, we calculate the magnitude of vector a:

||a|| = sqrt((5)^2 + (12)^2) = sqrt(25 + 144) = sqrt(169) = 13

Now we can compute the scalar projection:

scalar projection = rac{92}{13} ≈ 7.08

Step 2: Calculate the vector projection of b onto a.

To find the vector projection, we use the formula:

vector projection of b onto a = scalar projection × rac{a}{||a||}

We already have the scalar projection (≈ 7.08) and the unit vector of a:

unit vector of a = rac{a}{||a||} = (rac{5}{13}, rac{12}{13})

Now, we can compute the vector projection:

vector projection = 7.08 × (rac{5}{13}, rac{12}{13}) = (rac{35.4}{13}, rac{84.96}{13}) ≈ (2.73, 6.54)

Conclusion:

The scalar projection of vector b onto vector a is approximately 7.08, and the vector projection is approximately (2.73, 6.54).