vector PQ is parallel to vector v [-4, -6] and has initial point P(-7,5). Determine the coordinates of Q.
Assuming it is in the same direction, then if p is the vector to P, then
p+v = [-11,-1]
Q = (-11,-1)
Q could also be the end point of p-v.
Well, if vector PQ is parallel to vector v[-4, -6], it means that the direction of PQ is the same as v. We can find the coordinates of Q by moving in the same direction as v from the initial point P(-7, 5).
Now, let's follow vector v's direction: we move -4 units in the x-direction and -6 units in the y-direction. So, starting from P(-7,5), we move -4 units in the x-direction, which brings us to x = -7 - 4 = -11. Then, we move -6 units in the y-direction, which brings us to y = 5 - 6 = -1.
Therefore, the coordinates of Q are (-11, -1).
Now, let's hope that Q doesn't do anything too parallel to our sense of humor!
To determine the coordinates of point Q, we need to use the information given about vector PQ and its initial point P(-7, 5).
Given that vector PQ is parallel to vector v [-4, -6], we can conclude that the direction of vector PQ is the same as vector v. This means that the components of vector PQ will be a scalar multiple of the components of vector v.
Let's assume the scalar multiple is 'k'. Then, we can write the components of vector PQ as:
PQ = k * v
Since vector v is given as [-4, -6], the components of vector PQ will be:
PQ = [k * -4, k * -6]
Now, we need to find the value of 'k' to determine the magnitude of vector PQ. To do that, we can use the coordinates of point P(-7, 5) as the initial point of vector PQ.
PQ = [x-coordinate of Q - x-coordinate of P, y-coordinate of Q - y-coordinate of P]
Using the known values, we can write the equation:
[k * -4, k * -6] = [x-coordinate of Q - (-7), y-coordinate of Q - 5]
Simplifying this equation, we get:
[-4k, -6k] = [x-coordinate of Q + 7, y-coordinate of Q - 5]
Comparing the x-components, we have:
-4k = x-coordinate of Q + 7
Comparing the y-components, we have:
-6k = y-coordinate of Q - 5
Solving these two equations simultaneously will give us the values of the x-coordinate and y-coordinate of point Q.
From the first equation, we can isolate x-coordinate of Q:
x-coordinate of Q = -4k - 7
Substituting this value into the second equation, we get:
-6k = -4k - 7 + 5
Simplifying this equation, we have:
-6k = -4k - 2
Bringing -4k to the left side, we get:
-2k = -2
Dividing both sides by -2, we find:
k = 1
Therefore, the scalar multiple 'k' is 1.
Now, substituting the value of k = 1 into the equation for the x-coordinate of Q:
x-coordinate of Q = -4(1) - 7
= -4 - 7
= -11
Similarly, substituting the value of k = 1 into the equation for the y-coordinate of Q:
y-coordinate of Q = -6(1) + 5
= -6 + 5
= -1
Therefore, the coordinates of point Q are (-11, -1).
To determine the coordinates of point Q, which is the terminal point of vector PQ, we can use the following steps:
Step 1: Understand the problem and gather the given information.
We are given that vector PQ is parallel to vector v, which is [-4, -6]. We also know the initial point, P, which has coordinates (-7, 5).
Step 2: Recall the definition of a vector.
A vector is defined by its magnitude and direction. In this case, the direction is given by the parallel vector v, and the magnitude is not explicitly provided.
Step 3: Use the parallel vector to determine the coordinate change.
Since vector PQ is parallel to vector v, the coordinate change from P to Q will be a multiple of the coordinate change represented by vector v. In other words, for every change of -4 units in the x-coordinate, we will have a corresponding change of -6 units in the y-coordinate.
Step 4: Calculate the coordinate change.
To determine the coordinate change from P to Q, we multiply the x-coordinate change (-4) by the number of times it occurs and the y-coordinate change (-6) by the same number of times.
Step 5: Calculate the coordinates of point Q.
To calculate the x-coordinate of point Q, we add the coordinate change to the x-coordinate of point P. Similarly, to find the y-coordinate of point Q, we add the coordinate change to the y-coordinate of point P.
So, let's calculate the coordinate change:
Coordinate Change = (Number of times x-coordinate change, Number of times y-coordinate change)
Since we only have the parallel vector v, we don't have the magnitude of vector PQ or the exact number of times the coordinate change occurs. Therefore, the coordinate change can be represented as (-4m, -6m), where 'm' represents any non-zero real number.
Now, let's determine the coordinates of point Q:
x-coordinate of Q = x-coordinate of P + x-coordinate change
= -7 + -4m
= -7 - 4m
y-coordinate of Q = y-coordinate of P + y-coordinate change
= 5 + -6m
= 5 - 6m
Therefore, the coordinates of point Q are given by (-7 - 4m, 5 - 6m), where 'm' represents any non-zero real number.