Name: What Is Transformation of Axes?
Uploaded: 2019-07-19
Duration: 4 min 23 s
Description: Let's learn how the coordinates of a point change when we shift the coordinated axis by some distance. Watch this video and explore this concept of coordinat...

Question

If

O

is the origin and the co-ordinates of any two points

Q_{1}

and

Q_{2}

are

(x_{1}, y_{1})

and

(x_{2}, y_{2})

respectively, then prove that

O Q_{1} \cdot O Q_{2} \cos (∠ Q_{1} O Q_{2}) = x_{1} x_{2} + y_{1} y_{2}

Accepted Answer

We have given that If $O$ $(0, 0)$ is the origin and the co-ordinates of any two points $(x_{1}, y_{1})$ and $Q_{2}$ are $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ respectively.Now we should need to plot a diagram between $x - y$ plane or in co-ordinate system as following:

We also know that how can we find distance between two gives points :

$D = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} O Q_{1} = \sqrt{{(x_{1} - 0)}^{2} + (y_{1} - 0)^{2}} O Q_{1} = {x_{1}}^{2} + {y_{1}}^{2} O Q_{2} = {x_{2}}^{2} + {y_{2}}^{2} Q_{1} Q_{2} = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

By applying triangle rule:

${(Q_{1} Q_{2})}^{2} = {(O Q_{1})}^{2} + {(O Q_{2})}^{2} - 2 (O Q_{1}) . (O Q_{2}) \cos (∠ Q_{1} O Q_{2}) {(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} = {x_{1}}^{2} + {x_{2}}^{2} + {y_{1}}^{2} + {y_{2}}^{2} - 2 (O Q_{1}) . (O Q_{2}) \cos (∠ Q_{1} O Q_{2}) {x_{1}}^{2} + {x_{2}}^{2} + {y_{1}}^{2} + {y_{2}}^{2} - 2 x_{1} x_{2} - 2 y_{1} y_{2} = {x_{1}}^{2} + {x_{2}}^{2} + {y_{1}}^{2} + {y_{2}}^{2} - 2 (O Q_{1}) . (O Q_{2}) \cos (∠ Q_{1} O Q_{2}) - 2 x_{1} x_{2} - 2 y_{1} y_{2} = - 2 (O Q_{1}) . (O Q_{2}) \cos (∠ Q_{1} O Q_{2}) (O Q_{1}) . (O Q_{2}) \cos (∠ Q_{1} O Q_{2}) = x_{1} x_{2} + y_{1} y_{2}$

Hence, the correct answer.

If $O$ is the origin and the co-ordinates of any two points $Q_{1}$ and $Q_{2}$ are $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ respectively, then prove that
$O Q_{1} \cdot O Q_{2} \cos (∠ Q_{1} O Q_{2}) = x_{1} x_{2} + y_{1} y_{2}$

Important Questions on Coordinate Systems and Coordinates

Learn and Prepare for any exam you want !

If O is the origin and the co-ordinates of any two points Q1 and Q2 are x1,y1 and x2,y2 respectively, then prove that OQ1·OQ2cos∠Q1OQ2=x1x2+y1y2

Important Questions on Coordinate Systems and Coordinates

Learn and Prepare for any exam you want !

If $O$ is the origin and the co-ordinates of any two points $Q_{1}$ and $Q_{2}$ are $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ respectively, then prove that
$O Q_{1} \cdot O Q_{2} \cos (∠ Q_{1} O Q_{2}) = x_{1} x_{2} + y_{1} y_{2}$