Equivalent Vectors

Equivalent vectors: When two vectors have same magnitude and direction then such vectors are called vectors. The magnitude of the vectors can be found by using the distance formula. In $\left \| \vec{AB} \right \|$ if the co-ordinate of initial point A(x1,y1) and B(x2,y2) then magnitude of AB ,

$\sqrt{(x_{2}-x_{1})^2 +(y_{2}-y_{1})^2}$

Example on equivalent vectors

 1) Show that vectors PQ and RS are equivalent vectors. Solution: To show that the two vectors are equivalent, we will find the magnitude of each vectors. $\left \| \vec{PQ} \right \|=\sqrt{(4-1)^2 +(4 - 2)^2}$ = $\sqrt{13}$ $\left \| \vec{RS} \right \|=\sqrt{(3-0)^2 +(2 - 0)^2}$ = $\sqrt{13}$ Moreover both the line segments have same direction since both are directed upward. Both the vectors have same magnitude $\sqrt{13}$ from the distance formula. $\left \| \vec{PQ} \right \| = \left \| \vec{RS} \right \|$ are equivalent vectors.
2) If a vector AB with coordinates of endpoints (1, -2) and (4, -1) is equal to vector CD with coordinates of endpoints (a, -1) and (0, 0). Find the value of a?
Solution:
As the two vectors are equal,
So, $\left \| \vec{AB} \right \|= \left \| \vec{CD} \right \|$

$\sqrt{(1-4)^+(-2 +1)^2} = \sqrt{(a - 0)^2 + (-1 -0)^2}$

$\sqrt{(9+1)} = \sqrt{(a^{2} +1}$

10 = $a^{2}$ +1
9= $a^{2}$
a =$\pm$3

Practice on equivalent vectors

1) Given that two endpoints of two vectors are A(-1, 3), B(2, 4) and C(1, -2), D(4, -1). Prove that $\left \| \vec{AB} \right \|$ and $\left \| \vec{CD} \right \|$ are equal vectors?
2)Given that two endpoints of two vectors are P(-1, 4), Q(5, 2) and R(1, -2), S(4, -5). Check whether the two vectors $\left \| \vec{PQ} \right \|$ and $\left \| \vec{RS} \right \|$ are equal vectors?