In mathematics, vectors are a fundamental concept that goes beyond numbers. Suppose I ask, "How far is your home from school?" One possible response is " \(2\) kilometers." However, I can’t get to your home with just this information. I would also need to know the direction, whether it is east, west, northeast, or south. This combination of both distance and direction is what a vector represents.
Pilots use vectors to calculate the distance and direction they need to take inorder to travel from one location to another. In this section, we will explore how to use vectors and apply different operations on them.
On a piece of graph paper, draw a sketch that shows your path from point \(\text{A}\) to \(\text{B}\) and then from \(\text{B}\) to \(\text{C}\text{.}\)
Think of a way on how you can represent the movement from point \(\text{A}\) to \(\text{B}\) using notation, also from point \(\text{B}\) to \(\text{C}\text{.}\)
Suppose you now move in a reverse way from point \(\text{C}\) to point \(\text{A}\) following the same path. How would you represent that movement using notation?
Vector notation is a way of representing quanties that have both magnitude and direction. Vector \(\textbf{PQ}\) can be denoted as \(\underset{\sim}{PQ}\) or \(\overrightarrow{PQ}\) or \(\mathbf{PQ}\text{.}\)The magnitude of vector \(\textbf{PQ}\) is represented as \(|\textbf{PQ}|\text{.}\) In this case, we refer to \(P\) as initial point and \(Q\) as the terminal point.
Additionally, a vector can also be represented using a single small letter,such as \(\mathbf{a}\) or \(\underset{\sim}{\mathbf{a}}\text{.}\) In Figure 2.8.3 below, we can represent a vector from a point \(\text{P}\) to point \(Q\) as \(\overrightarrow{PQ} = \mathbf{a} = \underset{\sim}{\mathbf{a}}\)
Similarly, if the direction of the vector is reversed, from point \(\text{Q}\) to point \(\text{P}\) the vector is represented as \(\overrightarrow{QP} = \mathbf{-a} = -\underset{\sim}{\mathbf{a}}\)
Vector \(\mathbf{AB} \text{ and } \mathbf{DC}\) are equivalent because they have the same magnitude, \(|\mathbf{AB}| = |\mathbf{DC}| \text{,}\) and they point in the same direction
Write the resultant displacement in coordinate form \(\begin{pmatrix} x \\ y \end{pmatrix}\text{,}\) where \(x\) represents displacement along the \(x\) axis and \(y\) represents displacement along the \(y\) axis.
Consider a displacement from point \(P\) to point \(Q\text{,}\) followed by another displacement from point \(Q\) to point \(N\text{.}\) The total resultant displacement from \(P\) to \(N\) is obtained by adding the two vectors sequentially.
\(PQNM\) is a square with vectors \(\mathbf{PQ}\) and \(\mathbf{PM}\) given as \(\mathbf{a} \text{ and } \mathbf{b}\) respectively, as shown in Figure 2.8.12 . Express the \(\mathbf{PN}\) and \(\mathbf{MQ}\) vectors in terms of \(\mathbf{a}\) and \(\mathbf{b}\)
Given the vectors \(\mathbf{a} = \binom{2}{3}\) and \(\mathbf{b} = \binom{4}{-1}\text{,}\) find \(\mathbf{a} + \mathbf{b}\) and illustrate the solution graphically.
In Figure 2.8.14, the vector \(\mathbf{PQ}\) is represented as \(\mathbf{a}\text{.}\) When we multiply \(\mathbf{a}\) by a positive scalar, say \(2\text{,}\) the length of the vector doubles, making it \(\mathbf{2a}\) as shown in Figure 2.8.15. The direction of the vector remains unchanged, but its magnitude increases.
In Figure 2.8.15, the vector \(\mathbf{PN}\) is given by: \(\mathbf{PN} = \mathbf{a} + \mathbf{a} = 2\mathbf{a}\text{.}\) This means \(\mathbf{PN}\) has the same direction as \(\mathbf{PQ}\text{,}\) but its magnitude twice that of \(\mathbf{PQ}.\)
Consider the vector \(\mathbf{AB}\text{,}\)denoted as \(\mathbf{a}\text{,}\) in Figure 2.8.16 The vector points to the right and has a magnitude of \(\mathbf{a}\text{.}\)
When a vector \(\mathbf{a}\text{,}\) as shown in Figure 2.8.18, is multiplied by \(0\text{,}\) its magnitude becomes \(0\text{,}\) resulting in a zero vector.
A pentagon \(ABCDE\) with \(\overrightarrow{AB} = m\text{,}\)\(\overrightarrow{BC} = n\text{,}\) and \(\overrightarrow{CD} = k\text{.}\) Express the following vectors in terms of \(m\text{,}\)\(n\text{,}\) and \(k:\)
Represent vector AB in terms of its components as \(\begin{pmatrix} x \\ y \end{pmatrix}\) where \(x\) is the horizontal displacement and \(y\) is the vertical displacement.
A vector expressed in the form of \(\begin{pmatrix} a \\ b \end{pmatrix}\text{,}\) where \(a\) is the horizontal displacement along the \(x\) axis and \(\mathbf{b}\) is the vertical displacement along the \(y\) axis is known as a column vector.
The vector \(\textbf{OP}\) illustrates a displacement from the origin \(O(0,0)\) to the point \(P(4,5)\text{.}\) This consist of a horizontal displacement of \(4\) units along the \(x\) axis and a vertical displacement of \(5\) units in the \(y\) axis.
Given that: \(\mathbf{a} = \binom{1}{4}\) and \(\mathbf{b} = \binom{5}{3}\text{.}\) Find \(\mathbf{a} + \mathbf{b}\) and illustrate the solution graphically.
Begin at the point \((1,0)\) on the grid, move \(1\) unit horizontally to the right and move \(4\) units vertically upwards and mark it as end point. Draw a directed line connecting the two points as shown in the Figure 2.8.22.
From the point \((4,0)\) on the grid, move \(5\) units horizontally to the right parallel to the \(x\) axis, and move \(3\) units vertically up and mark it as end point. Draw another directed line to join the two points.
Now, to find the resultant vector \(\mathbf{a} + \mathbf{b}\text{,}\) join the initial point \((1,0)\) with the final point \((7,7)\) and count the total displacements in the \(x\) and \(y\) directions.
To determine \(2\mathbf{a} + 5\mathbf{b}\text{,}\) we multiply vector \(\mathbf{a}\) by \(2\) and vector \(\mathbf{b}\) by \(5\) and finally we add the resulting vectors.
Similarly, for point \(A\) in the plane its position vector \(\textbf{OA}\) is denoted by \(\mathbf{a}\text{.}\) Also for point \(B\) in the plane it’s position vector \(\textbf{OB}\) is denoted by \(\mathbf{b}\text{.}\)
From Point \(O\text{,}\) move \(3\) units to the right along the \(x\) axis and \(4\) units upward in the \(y\) axis. Mark this new position as Point \(A\text{.}\)
The magnitude of \(\overrightarrow{AB}\) in Figure 2.8.27 can be denoted as \(|\mathbf{AB}|\text{.}\) The magnitude of \(\overrightarrow{AB}\) represents the distance between point \(\text{A}\) and point \(\text{B}\text{.}\)
We can represent the components of \(\overrightarrow{AB}\) as \(\begin{pmatrix} x \\ y \end{pmatrix}\text{,}\) where \(x\) represents the horizontal displacement and \(y\) represents the vertical displacement.
The magnitude of a vector is always positive since \(x\) and \(y\) components are squared, resulting in \(x^2\) and \(y^2\text{,}\) both of which are non-negative.
From Point \(A\text{,}\) move \(6\) units to the right parallel to the \(x\) axis and mark this new location as Point \(B\text{.}\) Write down its coordinates.
Consider the cordinates of point \(P\) given as \((x_1,y_1)\) and point \(N\) given as \((x_2,y_2)\) and \(M\) is the midpoint of \(\mathbf{PN}\) as shown in figure below.
Consider triangle \(PQR\) where \(\overrightarrow{PQ} = \mathbf{u}\) and \(\overrightarrow{PR} = \mathbf{v}.\) Point \(S\) is located on \(PR\) such that \(PS:SR = 2:3\text{.}\) Point \(T\) lies on \(QR\) with \(QT:TR = 3:2\text{,}\) and point \(U\) is on \(PQ\) such that \(PU:UQ = 4:1\text{.}\) Determine the following vectors in terms of \(\mathbf{u}\) and \(\mathbf{v}\text{:}\)
Translate each point by moving \(2\) units to the right parallel to the \(x\) axis and \(3\) units up in the \(y\) axis. Label the new points as \(A', B',\) and \(C'\text{.}\)
Use dotted lines to connect each original point to its corresponding translated point \((A \text{ to } A', B \text{ to } B', C \text{ to } C')\text{,}\) add arrows to indicate the direction.
A square \(\mathbf{ABCD}\) undergoes a translation when each of its vertices (\(\mathbf{A}\text{,}\)\(\mathbf{B}\text{,}\)\(\mathbf{C}\) and \(\mathbf{D}\)) is moved the same distance and in the same direction. A translation vector, denoted by \(\mathbf{T}\text{,}\) describes this movement.
Using \(\mathbf{T}\) to represent a translation, the notation \(\mathbf{T}(\mathbf{P})\) indicates the application of the translation \(\mathbf{T}\) on \(\mathbf{P}\text{.}\) In Figure 2.8.33, shows \(\mathbf{A'B'C'D'}\) is the image of \(\mathbf{ABCD}\) under a translation.
Triangle \(\mathbf{ABC}\) has vertices \(\mathbf{A}(1,3)\text{,}\)\(\mathbf{B}(3,0)\) and \(\mathbf{C}(4,4)\text{.}\)The triangle undergoes a translation \(\mathbf{T}\) defined by the vector \(\begin{pmatrix} 4 \\ 3 \end{pmatrix}\text{.}\)
Draw triangle \(\mathbf{XYZ}\) with vertices \(\mathbf{X(1, 4)}\text{,}\)\(\mathbf{Y(6, 2)}\text{,}\) and \(\mathbf{Z(5, 3)}\text{.}\) On the same axes, plot \(\mathbf{X'Y'Z'}\text{,}\) the image of triangle \(\mathbf{XYZ}\) under a translation given by \(\binom{4}{9}.\)
A point \(\mathbf{P(5, -3)}\) is mapped to a new position after a translation. If the new coordinates are \(\mathbf{(9, 1)}\text{,}\) determine the translation vector used.
A point \(\mathbf{M(1, -4)}\) undergoes a translation by \(\binom{3}{5}.\) Determine the coordinates of \(\mathbf{M'}\text{,}\) the transformed point. If \(\mathbf{M'}\) is then translated by \(\binom{-4}{2},\) find the final position \(\mathbf{M''}\text{.}\) What is the single translation vector that maps \(\mathbf{M'}\) to \(\mathbf{M''}\) directly?
Given that \(a=\)\((-3,2)\) , \(b=\)\((6,-4)\) and \(c=\)\((5,-15) \)and that \(q=\)\(2a\)+ \(\frac{1}{2}b\) +\(\frac{1}{5}c\text{.}\) Express \(q\) as a column vector and hence calculate its magnitude \(|q|\) correct to two decimal places
If \(P\text{,}\)\(Q\) and \(R\) are the points \((2, - 4)\text{,}\)\((4, 0)\) and \((1, 6)\) respectively, use the vector method to find the coordinates of point \(S\) given that \(PQRS\) is a parallelogram.
In triangle \(OAB\text{,}\)\(M\) and \(N\) are points on \(OA\) and \(OB\) respectively, such that \(OM:OA=2:5\) and \(ON:OB=2:3\text{.}\)\(AN\) and \(BM\) intersect at \(T\text{.}\) Given that \(OA=a\) and \(OB=b\text{.}\)Express in terms of \(a\) and \(b\text{:}\)
