Collinear Vector Help: Finding the Value of p for Collinearity | Origin O

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In summary, the vector \displaystyle \overrightarrow{OA} has magnitude 100 and has the same direction as \displaystyle \left(\begin{array}{cc}7\\24\end{array}\right). The vector \displaystyle \overrightarrow{OB} has magnitude 24 and has the same direction as \displaystyle \left(\begin{array}{cc}24\\99\end{array}\right).
  • #1
omicron
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The position vectors of A, B and C relative to an origin O are [tex]-I+pj[/tex], [tex]5i+9j[/tex] & [tex]6i+8j[/tex] respectively. Determine the value of p for which A, B & C are collinear.
 
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  • #2
Have you learned any coordinate geometry at school?
 
  • #3
Yes I have.
 
  • #4
So you may instead assign the coordinates (-1,p) to the position vector -i + pj, (5,9) to the position vector 5i + 9j and (6,8) to the position vector 6i + 8j.

If three points are collinear, this means that the gradient between any two points of the three is the same.
 
  • #5
Thank you!
 
  • #6
One more question.
a) The vector [tex]\displaystyle \overrightarrow{OA}[/tex] has magnitude 100 and has the same direction as [tex]\displaystyle \left(\begin{array}{cc}7\\24\end{array}\right)[/tex]. Express [tex]\displaystyle \overrightarrow{OA}[/tex] as a column vector.
b) The vector [tex]\displaystyle \overrightarrow{OB}[/tex] is [tex]\displaystyle \left(\begin{array}{cc}24\\99\end{array}\right)[/tex]. Obtain the unit vector in the direction of [tex]\displaystyle \overrightarrow{AB}[/tex].
 
  • #7
omicron said:
One more question.
a) The vector [tex]\displaystyle \overrightarrow{OA}[/tex] has magnitude 100 and has the same direction as [tex]\displaystyle \left(\begin{array}{cc}7\\24\end{array}\right)[/tex]. Express [tex]\displaystyle \overrightarrow{OA}[/tex] as a column vector.
b) The vector [tex]\displaystyle \overrightarrow{OB}[/tex] is [tex]\displaystyle \left(\begin{array}{cc}24\\99\end{array}\right)[/tex]. Obtain the unit vector in the direction of [tex]\displaystyle \overrightarrow{AB}[/tex].

You need to show evidence of some work. Do you know what unit vectors are?
 
  • #8
Yes I do know.
 
  • #9
Then you should be able to solve both of those problems..
 
  • #10
If I did, I wouldn't have posted them. :bugeye:
 
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  • #11
omicron said:
One more question.
a) The vector [tex]\displaystyle \overrightarrow{OA}[/tex] has magnitude 100 and has the same direction as [tex]\displaystyle \left(\begin{array}{cc}7\\24\end{array}\right)[/tex]. Express [tex]\displaystyle \overrightarrow{OA}[/tex] as a column vector.

First of all, find the magnitude of the vector [tex]\displaystyle \left(\begin{array}{cc}7\\24\end{array}\right)[/tex]. What is it?

b) The vector [tex]\displaystyle \overrightarrow{OB}[/tex] is [tex]\displaystyle \left(\begin{array}{cc}24\\99\end{array}\right)[/tex]. Obtain the unit vector in the direction of [tex]\displaystyle \overrightarrow{AB}[/tex].

Do you know how to calculate the vector [tex]\displaystyle \overrightarrow{AB}[/tex]? (Hint: use information from a)
 

1. What is a collinear vector?

A collinear vector is a vector that lies on the same line as another vector. This means that they have the same direction and magnitude, but may have different starting points.

2. How do you determine if two vectors are collinear?

To determine if two vectors are collinear, you can use the collinearity condition which states that if two vectors are collinear, then one vector can be expressed as a scalar multiple of the other vector. This means that if you can multiply one vector by a constant and get the other vector, then they are collinear.

3. Can three or more vectors be collinear?

Yes, three or more vectors can be collinear if they all lie on the same line. This means that they all have the same direction and magnitude, but may have different starting points.

4. How do you calculate the collinear vector of a given vector?

The collinear vector of a given vector can be calculated by multiplying the given vector by a scalar. The scalar is determined by finding the ratio of the collinear vector's magnitude to the given vector's magnitude. This will result in a vector with the same direction as the given vector, but with a different magnitude.

5. What is the difference between a collinear vector and a parallel vector?

While both collinear and parallel vectors lie on the same line, the main difference is that collinear vectors have the same magnitude and direction, while parallel vectors can have different magnitudes but still have the same direction. Additionally, collinear vectors must have a starting point that lies on the same line, while parallel vectors do not necessarily need to have this starting point.

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