# Vectors: Collinear Demo - P,Q,R & D,E,F

• Macleef
In summary, a vector is a mathematical quantity with both magnitude and direction, often represented as an arrow. Vectors are collinear when they lie on the same line or are parallel to each other, and this can be determined using the cross product or dot product. In the context of collinear vectors, P, Q, and R represent vectors while D, E, and F represent points. Collinear vectors are commonly used in real-life applications, such as navigation and engineering, to calculate velocity and forces.
Macleef

## Homework Statement

Using vectors, demonstrate that these points are collinear.

a) P(15 , 10) , Q(6 , 4) , R(-12 , -8)

b) D(33, -5, 20) , E(6, 4, -16) , F(9, 3, -12)

## Homework Equations

$$\frac{x_{1}}{x_{2}}$$ = $$\frac{y_{1}}{y_{2}}$$

$$\frac{x_{1}}{x_{2}}$$ = $$\frac{y_{1}}{y_{2}}$$ = $$\frac{z_{1}}{z_{2}}$$

## The Attempt at a Solution

a)
Vector PQ = (-9 , -6)
Vector QR = (-18 , -12)
Vector RP = (27 , -18)

(-9 / 27 / -18) = (-6 / -18 / -12)

Therefore, not collinear.

b)
Vector DE = (-27, 9, -36)
Vector EF = (3, -1, 4)
Vector FD = (24, -8, 32)

(-27 / 3 / 24) = (9 / - 1 / -8) = (-36 / 4 / 32)
Am I correct or did I do something wrong? If I did, can you please point it out and tell me on how to fix it?

Last edited:
Since you don't say what x1, x2, y1, and y2 have to do with this problem you "relevant equation" really doesn't make sense.

The problem said "Usining vectors". Okay, what is the vector from P(15, 10) to Q(6,4)? What is the vector from P(15, 10) to R(-12, -8)?

well what u need to do is form two vectors like PQ as a vector, and QR, also as a vector. Where P is the starting point wheras Q is the end of the vector for the first one, and similarly for the second. I assume you are working on a cartesian system of coordinates. so now if you manage to show something similar to

PQ=k*QR, where k is a constant, than i guess you also have managed to show that those four points are collinear, since they all lie in a line!

Well, Halls is faster!

HallsofIvy said:
Since you don't say what x1, x2, y1, and y2 have to do with this problem you "relevant equation" really doesn't make sense.

The problem said "Usining vectors". Okay, what is the vector from P(15, 10) to Q(6,4)? What is the vector from P(15, 10) to R(-12, -8)?

I don't follow. . .

The question does mention what are x1, x2, y1, y2, (if relevant) z1 and z2. . .

And from my attempted solution, I did find the vectors. . .but I don't know what to do after finding the vectors. . .

why don't u just follow the suggestions! for the first part you have
PQ(-9,-6)
QR(-18,-12) ,now you see that their coordinates are proportional, that is

-9/-18=-6/-12=1/2.

Just do the same thing with the other!
Also you can proceede with PR like halls suggested, you will get the same thing.

## 1. What is a vector?

A vector is a mathematical quantity that has both magnitude and direction. It is often represented visually as an arrow, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction of the vector.

## 2. What does it mean for vectors to be collinear?

Vectors are collinear when they lie on the same line or are parallel to each other. This means that they have the same direction or opposite directions, but may have different magnitudes.

## 3. How can you determine if vectors are collinear?

To determine if vectors are collinear, you can use the cross product or dot product. If the cross product of two vectors is equal to zero, then they are collinear. Alternatively, if the dot product of two vectors is equal to the product of their magnitudes, then they are collinear.

## 4. What is the significance of P, Q, R & D, E, F in the context of collinear vectors?

P, Q, and R are typically used to represent vectors, while D, E, and F are used to represent points. In the context of collinear vectors, these letters are often used to label the vectors or points to show their relationship and determine if they are collinear.

## 5. How are collinear vectors used in real-life applications?

Collinear vectors are used in many real-life applications, such as navigation, physics, and engineering. For example, in navigation, collinear vectors can be used to determine the direction and magnitude of a plane's velocity, while in engineering, they can be used to calculate the forces acting on a structure.

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