Part A
Given that P=(5,4), Q=(7,3), R=(8,6), and S=(4,1), find the component form of the vector PQ+3RS.
a. <-10,-16>****
b. <-6,4>
c. <-2,-6>
d. <14,14>
Part B
Use the information from Part A to find the magnitude of the vector PQ+3RS.
a. 356
b. sqrt(26)
c. 2*sqrt(10)
d. 2*sqrt(89)
I know that Part A is correct, but I do not understand how to find the magnitude.
a. PQ=(7-5)x+(3-4(y)=2 ,-1
RS=(4-8)x+(1-6)y=-4,-5
PQ+3RS=2 ,-1 +3( -4,-5)=(2-14,-1-15)=10,-16
magnitude= sqrt(10^2+(-16)^2)=sqrt(356)
factoring 356
2*178
2*2*89
magnitude=2sqrt(89)
In general if you have points P(a,b) and Q(c,d)
then vector PQ = <c-a , d-b>
Notice I went from P to Q, so the subtraction is "destination x - starting x" , same for the y
Then to get vector QP we would get <a-c, b-d>
for yours, PQ = <7-5, 3-4> = <2,-1>
to find the magnitude of vector <x,y> find √( x^2 + y^2)
e.g. |PQ| = √(2^2 + (-1)^2 ) = √5
since your part a) is correct simply follow the above rule, I can see the answer
I would also appreciate if someone could better explain how to get Part A because my teacher did not explain this very well.
Well, don't worry, I'll be your guide through this mathematical circus! Let's start with Part A.
To find the component form of the vector PQ, we subtract the coordinates of Q from the coordinates of P:
PQ = (7-5, 3-4) = (2, -1)
Now let's find the component form of the vector RS:
RS = (4-8, 1-6) = (-4, -5)
To find the vector PQ+3RS, we multiply the coordinates of RS by 3 and add the coordinates of PQ:
PQ + 3RS = (2, -1) + 3(-4, -5) = (2, -1) + (-12, -15) = (-10, -16)
So the component form of the vector PQ+3RS is <-10, -16>. Your answer choice a. is correct!
Now, let's move on to Part B, finding the magnitude of the vector PQ+3RS.
The magnitude of a vector can be found using the formula: magnitude = sqrt(x^2 + y^2), where x and y are the components of the vector.
For the vector <-10, -16>, the magnitude is:
sqrt((-10)^2 + (-16)^2) = sqrt(100 + 256) = sqrt(356)
So, the magnitude of the vector PQ+3RS is sqrt(356). Your answer choice a. is correct!
Now, wasn't that a circus of calculations? I hope I've cleared things up for you.
To find the component form of a vector PQ+3RS, we first need to calculate the vector PQ and the vector RS separately.
Vector PQ can be found by subtracting the coordinates of point P from the coordinates of point Q:
PQ = Q - P = (7 - 5, 3 - 4) = (2, -1)
Similarly, vector RS can be found by subtracting the coordinates of point S from the coordinates of point R:
RS = S - R = (4 - 8, 1 - 6) = (-4, -5)
Now, to find the vector PQ+3RS, we can add 3 times the vector RS to the vector PQ:
PQ+3RS = (2, -1) + 3(-4, -5) = (2, -1) + (-12, -15) = (2 - 12, -1 - 15) = (-10, -16)
So, the correct answer for Part A is option a. <-10,-16>.
Part B - Finding the magnitude of a vector
To find the magnitude of a vector, we use the formula:
Magnitude = √(x^2 + y^2)
For the vector PQ+3RS, the x-component is -10 and the y-component is -16.
Magnitude = √((-10)^2 + (-16)^2) = √(100 + 256) = √356 = 2√89
Therefore, the correct answer for Part B is option d. 2√89.