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Problem 4 Easy Difficulty

Square $A B C D$ is drawn as shown below with the diagonals intersecting at $E$ a. State four pairs of equivalent vectors.
b. State four pairs of opposite vectors.
c. State two pairs of vectors whose magnitudes are equal but whose directions are perpendicular to each other.

Answer

a. $\overrightarrow{A D}=\overrightarrow{B C} ; \overrightarrow{A B}=\overrightarrow{D C} ; \overrightarrow{A E}=\overrightarrow{E C}$
$\overrightarrow{D E}=\overrightarrow{E B}$
b. $\overrightarrow{A D}=-\overrightarrow{C B} ; \overrightarrow{A B}=-\overrightarrow{C D}$
$\overrightarrow{A E}=-\overrightarrow{C E} ; \overrightarrow{E D}=-\overrightarrow{E B}$
$\overrightarrow{D A}=-\overrightarrow{B C}$
c. $\overrightarrow{A C} \& \overrightarrow{D B} ; \overrightarrow{A E} \& \overrightarrow{E B} ; \overrightarrow{E C} \& \overrightarrow{D E}$
$\overrightarrow{A B} \& \overrightarrow{C B}$

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Video Transcript

And this problem a figure is given which is shown over here. This figure shows a square baby, C. B. And the diagonals intersect at the point E. So that means that these two b. d. and easier the diagonals and the diagnose of a square intersected with at a 90° angle they intercepted right angle. So this angle will be a 90° angle. And also the diagonals bisect each other in the case of a square. So A C and B. D are bisected at the point. Now, first of all we need to state four pairs of equivalent vectors are equal victims. That is to victor's which have the same magnitude and the same direction. Now first of all notice that one of the pairs is A. D. Which is equal or equivalent to the victor B. C. This is because A. D. And B. C will have the same magnitude because they are both sides of the square. So their lengths will be the same as the magnitude will be the same. Also the direction of A. D. And B. C. And both seem to be the same. So they are equal vectors similarly, E. B will be equal to the vector dc because once again they are both sides of the square. They have the same land and hence the same magnitude and the directions are also the same. Also, we will have E. Is equal to E. C. This is because A. And E. C. Are both in the same direction. Also, since the diagonals are bisected at the point E. So the length of E will be equal to the length of the sea and hence the magnitudes will be the same. Also, we will have D. E is equal to E. V. And the reason for that is the same. He will be the point of the midpoint of B. D. So D. E and E. V are in the same direction and the length of D is equal to the length of E. V. Since the diagnosis by sector and E. Hence the magnitudes are the same and the D E is equal to maybe next. We need to stay four pairs of opposite vectors, so we will have A. B is equal to minus C. B. So A. B and C. B will be opposite vectors. We can obtain this from the first relation over here, A. D is equal to B. C. So that means B. C is equal to minus C. B. So the direction of CB is opposite the direction of A. D. And thus they will be opposite vectors because their magnitudes are the same, but the directions are opposite. Similarly, we have A. B is equal to D. C. So we will have A. B. Is equal to minus C. D. Also sends A is equal to easy, we'll have E. Is equal to minus C. E. We also have the E. Is equal to E. V. So we will have the E. Is equal to minus B. E. T. Now, in all of these cases we will have opposite vectors. So the opposite vectors will be A. D. And C. D. E. B. And C. D. E. N C E B, E N B. E. Next. Next we need to stay two pairs of vectors whose magnitudes are equal but whose directions are perpendicular to each other. So for that consider E and E. B. The victors E N E B. From this diagram, we can see that E. V are at a 90 degree angle, so they are perpendicular to each other. Also since the length of the diagnosed in a square and equal and is the midpoint of the diagnosed. Since the length of E will be equal to the length of E. B. The magnitude will be the same, and thus we have the two required victors of the same magnitude, but the directions are perpendicular to each other. Similarly, we can consider E. C. And E. They are one second perpendicular to each other and for the same reason the diagnose are equal in a square and E by six. The diagnosed, so the lens of the lens of E. C. And D. E will be the same. Southern magnitudes will be the same, but the directions are properly color to each other, so the other pair will be easy and GT and that is all that we have been asked to find.