Vectors: 7 Examples
In this course, we will go through 7 selected examples that demonstrate how to apply the key concepts learned in vectors. Each example will focus on different skills, such as vector representation, addition and subtraction, scalar multiplication, and solving geometric problems using position vectors.
Related Lessons:
Example 1
Given \(\vec{OA}=\left(\begin{matrix}-1\\2\end{matrix}\right)\), \(\vec{OB}=\left(\begin{matrix}1\\4\end{matrix}\right)\), find
- the coordinates of C if \(\vec{OC}=3\vec{OB}-\vec{OA}\)
- the values of m and n if \(\vec{OD}=m\vec{OA}+n\vec{OB}\) and D is the point (5,2)
- the vector \(\vec{AD}\) and \(\left|\vec{AD}\right|\)
Example 2
In the diagram below, the points a and b have position vectors and with respect to O. Given that two other points P and Q are such that \(\vec{QP}=kb\) and \(\vec{OQ}=ha\)
- Mark and label clearly the point Q
- Calculate the value of h and k.
Example 3
Given that \(\vec{AB}=\left(\begin{matrix}5\\-3\end{matrix}\right)\), \(\vec{AD}=\left(\begin{matrix}-1\\1\end{matrix}\right)\), and D is the point \((2,-1)\)
- Express \(\vec{BD}\) as a column vector
- Find the coordinates of A
Example 4
OXYZ is a parallelogram such that \(\vec{XY}=\left(\begin{matrix}3\\5\end{matrix}\right)\) and X is \((5,1)\).
- Express \(\vec{ZX}\) as a column vector
- The point P lies on ZX produced and \(\vec{XP}=k\vec{ZX}\). Show that \(\vec{OP}=\left(\begin{matrix}5+2k\\1-4k\end{matrix}\right)\)
- Given that P lies on the x-axis, find the value of k.
Example 5
The diagram above shows vector \(x\) and \(y\) and the points P and A. Draw and label clearly on the given diagram the points B and R such that:
- \(\vec{PR}=x+2y\)
- \(\vec{AB}=\frac{3}{2}y-x\)
Example 6
In the diagram below, ST is parallel to RQ whereby \(\frac{ST}{RQ}=\frac{3}{4}\), \(\vec{PS}=s\), \(\vec{PT}=t\)
- Find (i) \(\vec{ST}\) (ii) \(\vec{QR}\)
- If P is the point (7,10) and T is the point (11,2), find column vector \(\vec{TP}\)
Example 7
In the diagram, \(\vec{BC}=4\vec{BD}\) and \(\vec{DA}=5\vec{DX}\). E is the midpoint of AC. \(\vec{BD}=a\) and \(\vec{CE}=b\).
- Express, as simply as possible, in terms of \(a\) and \(b\) i) \(\vec{DC}\) ii \(\vec{DA}\) iii) \(\vec{BE}\)
- Show that \(\vec{BX}=\frac{2}{5}\left(4a+b\right)\)