Let 𝑣⃗ be the vector from initial point 𝑃 to terminal point 𝑄. 𝑃(2,3) and 𝑄(−5,1)

Question

Let 𝑣⃗ be the vector from initial point 𝑃 to terminal point 𝑄. 𝑃(2,3) and 𝑄(−5,1)

a) Write 𝑣⃗ in terms of 𝑖⃗ and 𝑗⃗.

b) Find the unit vector that has the same direction as vector 𝑣⃗.

Hello! I need help solving these two questions overall pleaseeee, thank you!

Answer 1

(a)

First, remember that i and j are direction unit vectors. The x component of a vector is multiplied by i and the y component is multiplied by j. To find each component, we need to find the change in position from P to Q. In order to find the change in a value, remember final minus initial. In other words, the change in x would be x₂ - x₁ and the change in y would be y₂ - y₁. Here, Q is the second point and P is the first point, so we get

x = -5 - 2 = -7

y = 1 - 3 = -2

So, our vector would be v = (-7)i + (-2)j

(b)

A unit vector is just a vector with a magnitude of 1. To make a unit vector, all you have to do is divide the original vector by its magnitude. So, let's start with the magnitude:

The magnitude of a vector is √(x² + y²). To help you remember it, this equation comes from the Pythagorean theorem for right triangles! So, for our vector, the magnitude would be

√[ (-7)² + (-2)² ] = √53

Then, the unit vector of our vector would be v/√53 :

unit vector = (-7/ √53 )i + (-2/√53)j

To double check, you can use the magnitude equation on the unit vector. If you've done everything correctly, it should give you a value of 1!

√[ (-7/√53)² + (-2/√53)² ] = √(49/53 + 4/53) = √(53/53) = √1 = 1