Let u = <-5, -3>, v = <-6, -1>. Find -3u + 5v

Question

Let u = <-5, -3>, v = <-6, -1>. Find -3u + 5v

my answer : <-15, 4>

Find a • b.

a = 5i + 7j, b = -4i + 3j

my answer:<-20, 21>



Answer 1

1) Let u = <-5, -3>, v = <-6, -1>. Find -3u + 5v

# -3 x u = < -3 x (-5), -3 x (-3) > = < 15, 9 >

# 5 x v = < 5 x -6, 5 x -1 > = < -30, -5 >

# -3u + 5v = < 15 + (-30), 9 + (-5) > = < -15, 4 > ---> You're right!

I'm assuming you're finding the dot product?

2) Find a • b.

a = 5i + 7j, b = -4i + 3j

# a can be expressed as < 5, 7 > and b can be expressed as < -4, 3 >

# (5 x -4) + (7 x 3) = -20 + 21 = 1

I hope this helps! :)



Answer 2

Finding the scalar of a vector (multiplying a vector by a constant) is quite simple.

Given a vector <a, b>, we can calculate a scalar of this vector, let's call it r<a, b> using the following:

r<a, b> = <ra, rb>

Using this knowledge, we can calculate -3u + 5v based on the following

-3u + 5v = -3*<-5, -3> + 5*<-6, -1> = <15, 9> + <-30, -5> = <-15, 4>

Your answer is correct!

When we have vectors that are given in the format of a = 5i + 7j, we can rewrite these as a = <5, 7>.

Using this fact, we can rewrite a and b as the following:

a = <5, 7>

b = <-4, 3>

The dot product between two vectors is given by the following formula:

<u1, v1>.<u2, v2> = u1*u2 + v1*v2

Using this equation, we can calculate the dot product of a and b as follows:

a • b = <5, 7> • <-4, 3> = 5 * (-4) + 7 * 3 = -20 + 21 = 1. Therefore, a • b = 1.

You expressed your answer as a vector, but remember that dot products are scalars, which means that they are actual values and not vectors.