Find the scalar and vector projections of b onto a.

a = <3, −4> b = < 6, 0>

compab
projab

by definition,

the scalar projection of b onto a
is b•a/|a|
= (18+0)/5 = 18/5

If I recall the vector projection of b on a is simply the unit vector a/|a| times the scalar projection of b onto a

so a/|a| = [3/5, -4/5]
so finally the vector projection would be
(18/5)[3/5 , -4/5]
= [54/25 , -72/25] which is a vector

To find the scalar and vector projections of b onto a, we can use the following formulas:

The scalar projection of b onto a, compab, is given by:
compab = (b · a) / |a|
where (b · a) represents the dot product of vectors b and a, and |a| represents the magnitude of vector a.

The vector projection of b onto a, projab, is given by:
projab = (compab * a) / |a|

Let's calculate the scalar projection first:

a · b = (3 * 6) + (-4 * 0) = 18
|a| = √(3^2 + (-4)^2) = 5

compab = (18) / (5) = 3.6

Now, let's calculate the vector projection:

projab = (3.6 * <3, -4>) / 5
= <(3.6 * 3) / 5, (3.6 * -4) / 5>
= <10.8 / 5, -14.4 / 5>
= <2.16, -2.88>

Therefore, the scalar projection of b onto a is compab = 3.6, and the vector projection of b onto a is projab = <2.16, -2.88>.

To find the scalar projection (compab) and vector projection (projab) of b onto a, we will follow these steps:

Step 1: Calculate the dot product of vectors a and b
The dot product of two vectors a = <a1, a2> and b = <b1, b2> is given by the formula: a · b = (a1)(b1) + (a2)(b2). In this case, a = <3, -4> and b = <6, 0>. So, the dot product is calculated as follows:

a · b = (3)(6) + (-4)(0) = 18 + 0 = 18

Step 2: Calculate the magnitude of vector a
The magnitude of a vector a = <a1, a2> is calculated using the formula: |a| = sqrt(a1^2 + a2^2). In this case, a = <3, -4>. So, the magnitude is calculated as follows:

|a| = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5

Step 3: Calculate the scalar projection (compab) of b onto a
The scalar projection of b onto a is calculated using the formula: compab = (a · b) / |a|. In this case, a · b = 18 and |a| = 5. So, the scalar projection is calculated as follows:

compab = (a · b) / |a| = 18 / 5 = 3.6

Step 4: Calculate the vector projection (projab) of b onto a
The vector projection of b onto a is calculated using the formula: projab = (compab / |a|) * a. In this case, compab = 3.6 and a = <3, -4>. So, the vector projection is calculated as follows:

projab = (3.6 / 5) * <3, -4>
= (0.72) * <3, -4>
= <2.16, -2.88>

Therefore, the scalar projection (compab) of b onto a is 3.6 and the vector projection (projab) of b onto a is <2.16, -2.88>.