5. Vectors

a. Definition, Magnitude and Direction

2. Displacements

Displacement Vector
Vectors are often drawn between two points in order to show a change of position or displacement. In particular, if something moves from point P=(p1,p2)P=(p_1,p_2) to a point Q=(q1,q2)Q=(q_1,q_2), this change in position is commonly denoted by the displacement vector: PQ→=Q−P=⟨q1−p1,q2−p2⟩ \overrightarrow{PQ}=Q-P=\left\langle q_1-p_1,q_2-p_2\right\rangle Here, PP is the tail of the vector while QQ is the tip of the vector.

Suppose you trace a line segment which starts at the point A=(3,−2)A=(3,-2) and ends at the point B=(5,6)B=(5,6). What displacement vector describes this motion?

Subtract the coordinates of the initial position from those of the final position. The resulting displacement vector is AB→=B−A=(5,6)−(3,−2)=⟨2,8⟩ \overrightarrow{AB}=B-A=(5,6)-(3,-2)=\left\langle2,8\right\rangle The picture gives a graphical representation of this vector.

eg_vec3-256

Find the displacement vector from P=(1,3)P=(1,3) to Q=(4,7)Q=(4,7). Plot it.

Answer

PQ→=⟨3,4⟩\overrightarrow{PQ}=\left\langle3,4\right\rangle

ex_vec1347
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Solution

The displacement vector is: PQ→=Q−P=(4,7)−(1,3)=⟨3,4⟩ \overrightarrow{PQ}=Q-P=(4,7)-(1,3) =\left\langle3,4\right\rangle We plot the points P=(1,3)P=(1,3) and Q=(4,7)Q=(4,7) and connect the dots.

ex_vec1347
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