Vectors
A vector describes a movement from one point to another. A vector quantity has both Information to give the direction of travel, or the direction of a force, for example, a speed of 20 m s-1 to the left, or a force of 15 N to the right. and The magnitude tells us the size of the vector. (size).
A A quantity that requires only a size, for example, distance travelled is 20 m. quantity has only magnitude.
A vector can be represented by a line segment labelled with an arrow.
A vector between two points A and B is described as: \(\overrightarrow{AB}\), \(\mathbf{a}\) or \(\underline{a}\).
The vector can also be represented by the column vector \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\). The top number is how many to move in the positive \(x\)-direction and the bottom number is how many to move in the positive \(y\)-direction.
Vectors are equal if they have the same magnitude and direction regardless of where they are.
\(\overrightarrow{CD} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}\)
\(\overrightarrow{EF} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}\)
So \(\overrightarrow{CD} = \overrightarrow{EF}\)
A negative vector has the same magnitude but the opposite direction.
Vector \(\mathbf{-k}\) is the same as travelling backwards down the vector \(\mathbf{k}\).
Example
Triangle ABC is isosceles. X is the midpoint of AB, Y is the midpoint of BC and Z is the midpoint of AC.
Write, in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\), the vectors \(\overrightarrow{ZY}\), \(\overrightarrow{YC}\), \(\overrightarrow{ZA}\) and \(\overrightarrow{BX}\).
\(\overrightarrow{ZY} = \mathbf{a}\)
\(\overrightarrow{ZY}\) and \(\overrightarrow{AX}\) are equal vectors, they have the same magnitude and direction.
\(\overrightarrow{YC} = \mathbf{b}\)
\(\overrightarrow{YC}\) and \(\overrightarrow{XZ}\) are equal vectors, they have the same magnitude and direction.
\(\overrightarrow{ZA} = \mathbf{-c}\)
\(\overrightarrow{ZA}\) has the same magnitude as \(\overrightarrow{AZ}\) but the opposite direction.
\(\overrightarrow{BX} = \mathbf{-a}\)
\(\overrightarrow{BX}\) has the same magnitude as \(\overrightarrow{AX}\) but the opposite direction.