3.13.2 Vector problem solving
3.13.2 Vector Problem Solving
3.13.2 Vector problem solving
Parallel vectors
Parallel vectors are two or more vectors that have the same or opposite direction but
may have different magnitudes. In other words, if two vectors are parallel, they point in the same direction, even if they may be different in length or magnitude.
For example, consider two vectors \( u \) and \( v \) in three-dimensional space.
If \( u \) and \( v \) are parallel, then \( u = k\times v \) for some scalar \( k \), where \( k \) can be positive or negative.
This means that \( u \) and \( v \) have the same direction but may have different lengths.
If \( k \) is positive, then \( u \) and \( v \) point in the same direction, and if \( k \) is negative, then they point in opposite directions.
Parallel vectors are useful in physics, engineering, and other fields where vector analysis is important.
For example, in physics, the force and velocity vectors of a moving object can be parallel, and in engineering, the direction of two forces acting on a structure can be parallel.
Points on a straight line
Points that lie on a straight line are said to be collinear.
In other words, if three or more points lie on the same line, then they are collinear.
To show that three points A, B, and C are parallel, show that the vectors connecting the three points are parallel.
Vectors divided in ratios
If a point divides a line segment in the ratio \( p \) : \( q \), then:
\( AX = \frac{p}{p+q} AB \) and \( XB = \frac{q}{p+q} AB \)
Worked example:
In the diagram, \( OAB \) and \( OED \) are straight lines.
\( O \) is the origin, \( A \) is the midpoint of \( OB \) and \( E \) is the midpoint of \( AC \).
\( AC=a \) and \( CB=b \)
Find, in terms of \( a \) and \( b \), in its simplest form.
\( AB \)
The path from \( A \) to \( B \) is
\( AB=AC+CB \)
\( AB=a+b \)
\( OE \)
The path from \( O \) to \( E \) is
\( OE=OA+AE \)
\( A \) is the midpoint of \( OB \), So
\( OA=AB=a+b \)
\( E \) is the midpoint of \( AC \), So
\( AE=\frac{1}{2}AC=12a \)
\( OE=(a+b)+\frac{1}{2}a \)
\( OE=\frac{3}{2}a+b \)
the position vector of \( D \)
The path from \( O \) to \( D \) is
\( OD=OE+ED \)
From part (ii)
\( OE=\frac{3}{2}a+b \)
\( E \) is the midpoint of \( AC \) , So
\( EC=\frac{1}{2}a \)
Therefore \( ED \) must be equal to \( \frac{1}{2}a+xb \), where \( x \) is a constant to be found.
\( ED \) is parallel to \( OE \), So
\( ED=kOE \)
\( (\frac{1}{2}a+xb) = k(\frac{3}{2}a+b) \)
Compare the coefficients of \( a \) and \( b \)
\( \frac{1}{2} = \frac{3}{2}k \), \( x=k \)
\( \frac{1}{3}=k \), \( x=\frac{1}{3} \)
So
\( ED=\frac{1}{2}a+\frac{1}{3}b \)
Now substitute the values in \( OD=OE+ED \)
\( OD=(\frac{3}{2}a+b)+(\frac{1}{2}a+\frac{1}{3}b \)
\( OD=\frac{3+1}{2}a+\frac{3+1}{3}b \)
\( OD=2a+\frac{4}{3}b \)
So position vector of \( D \) is \( 2a+\frac{4}{3}b \)
Test yourself
Question 1:
\( AB=\binom {6} {-1} \) , \( BC=\binom{-2}{5} \), \( DC=\binom{2}{-3} \)
Find
\( AC \)
[2]
\( BD \)
[2]
\( |BC| \)
[2]
Question 2:
\( A \) is the point \( (4, 1) \) and \( AB=\binom{-3}{1} \).
Find the coordinates of \( B \).
[1]
Question 3:
\( OAB \) is a triangle and \( ABC \) and \( PQC\) are straight lines.
\( P \) is the midpoint of \( OA \), \( Q \) is the midpoint of \( PC \) and \( OQ\) : \( OB=3\) : \( 1 \).
\( OA=4a \) and \( OB=8b \)
Find, in terms of \( a \) and \( b \), in its simplest form.
\( AB \)
[1]
\( OQ \)
[1]
\( PQ \)
[1]
By using vectors, find the ratio \( AB \) : \( BC \)
[3]
Question 4:
\( OPTR \) is a trapezium
\( OP=a \)
\( PT=b \)
\( OR=3b \)
Find \( OT \) in terms of \( a \) and \( b \)
[1]
Find \( PR \) in terms of \( a \) and \( b \)
Give your answer in its simplest form.
[1]
\( S \) is the point on \( PR \) such that \( PS \) : \( SR=1 \) : \( 3 \)
Find \( OS \) in terms of \( a \) and \( b \)
Give your answer in its simplest form.
[2]
Question 5:
\( CE=kCD \)
Find the value of \( k \).
[1]
\( OAB \) is a triangle and \( C \) is the midpoint of \( OB \).
\( D \) is on \( AB \) such that \( AD \) : \( DB=3 \) : \( 5 \).
\( OAE \) is a straight line such that \( OA \) : \( AE=2 \) : \( 3 \).
\( OA=a \) and \( OC=c \)
Find the following vectors, in terms of \( a \) and \( c \) , in their simplest form.
\( AB \)
[1]
\( AD \)
[1]
\( CE \)
[1]
\( CD \)
[2]