My question is based on vectors....Its an MCQ typo.. I am new to vectors .... so not sure about the answer... pls answer with explanation

Question

Let vector C = Vector A + Vector B then
A) Magnitude is C always > Magnitude of A
B) It is possible to have Magnitude of C < Magnitude of A and Magnitude of C < Magnitude of B.
C) C is always equal to A+B
D) C is never equal to A+B

Answer 1

Vectors (as you probably know) are line segments with magnitude and direction. The magnitude of a vector is its length. I can write a vector like I would a point: v = (2,3). This means that my vector starts at the origin and goes to the point (2,3). That gives it magnitude (the length of the line segment connecting (0,0) and (2,3)) and direction.

 

Adding vectors means taking two vectors and basically attaching the second one's starting point to the first one's ending point. If v = (2,3) and w = (1,-2), their sum can be represented by drawing a line from the origin to (2,3) and then drawing a line from (2,3) that goes (right 1, down 2), like w does. v + w has its end point at (2+1, 3-2) = (3,1).

 

Now, to your question.

 

 

(A) FALSE

 

Here's a counter-example:

 

A = (3,0); B = (-2,0)

 

If I go three right along the x-axis (A) and then two left along the x-axis (B), I wind up at (1,0), only one point away from the origin. A's length is 3, B's length is 2, and C's length is 1. So, A's magnitude is greater than C's.

 

(B) TRUE

 

The example I just gave is a case where this happens. A's length is 3, B's length is 2, and C's length is 1, so C's magnitude is less than both A's and B's.

 

(C) TRUE

 

This is how you compute the sum of two vectors. So, by definition, C is equal to A + B. I did this above, with v and w. If A = (2,3) and B = (1,-2), A + B = C = (2+1, 3+(-2)) = (3,1). You can see this working by drawing a line from the origin to (2,3) and another line from the origin to (1,-2). Now, imagine moving the second line's start point from the origin to the first line's end point at (2,3). Where did you move the second line's end point to? That point defines C, and it had better be (3,1).

 

(D) FALSE

 

This one is false for the same reason (C) is true.

 

 

Hope this makes sense! Let me know if you need any clarification.