Question Video: Geometric Applications of Vectors in a Rectangle Mathematics

Consider the given figure. If π΄π΅πΆπ· is a rectangle and πΈ is the midpoint of the line segment π΄π·, then ππ + ππ β ππ = οΌΏ. [A] ππ [B] ππ [C] ππ [D] ππ

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Video Transcript

Consider the given figure. If π΄π΅πΆπ· is a rectangle and πΈ is the midpoint of the line segment π΄π·, then vector ππ plus vector ππ minus vector ππ is equal to blank. Is it (A) vector ππ, (B) vector ππ, option (C) vector ππ, or option (D) vector ππ?

We begin by recalling that two vectors are equal if they have the same direction and magnitude. This means that in a rectangle, the parallel sides will be represented by the same vectors as they have the same magnitude and direction. This means that vector ππ is equal to vector ππ and vector ππ is equal to vector ππ. We are also told that πΈ is the midpoint of the line segment π΄π·. This means that vector ππ is equal to vector ππ, and both of these are equal to one-half multiplied by vector ππ.

Letβs now consider the expression we are given, vector ππ plus vector ππ minus vector ππ. We know that the general vectors ππ and ππ are the additive inverse of each other. This means that vector ππ plus vector ππ is equal to the zero vector. It also means that the vector ππ is equal to the negative of vector ππ. This means that instead of subtracting vector ππ, we can add vector ππ. Our expression can be rewritten as ππ plus ππ plus ππ.

From the diagram, we can see that the vector ππ can be rewritten as vector ππ plus vector ππ. This gives us the expression ππ plus ππ plus ππ plus ππ. As vector ππ and vector ππ are the additive inverse of one another, they will sum to give us the zero vector. So, we can therefore cancel these terms. As vector ππ is equal to vector ππ, then vector ππ will be equal to vector ππ. Our expression can be simplified to vector ππ plus vector ππ.

We can change the order of these two vectors so that the endpoint of the first vector is the start point of the second vector. We can see from the diagram that the vector ππ plus the vector ππ is equivalent to the vector ππ. The correct answer is therefore option (B).

If π΄π΅πΆπ· is a rectangle and πΈ is the midpoint of line segment π΄π·, then the vector ππ plus the vector ππ minus the vector ππ is equal to the vector ππ.