Consider the vector 𝑣⎯⎯⎯ v ¯ between (−2,4,−5) ( − 2 , 4 , − 5 ) and (5,−2,−4) ( 5 , − 2 , − 4 ) .

Question

1. the vector v ¯ is?
2. THE LENGTH OF v ¯ IS?
3. if the tail of v ¯ is at (5,-1,1), then the tip is at?
4. if the tip of v ¯ is at (0, -5, -2) then the tail is at ?
5. what vector has the same length as v ¯ but points in the opposite direction?
6. What vector has the same direction as v ¯ but twice as long?

Answer 1

Answer:

Let A ( -2, 4, -5), B( 5,-2, -4)

1- The Vector ^→V = ^→A B = ( 5 -(-2)) i + (-2-4) j + (-4-(-5)) k = 7i - 6j + k

2- Length of ^→V = sqrt[( 7)² + (-6)² + ( 1)²] = √86 = 9.3

3- if the tail of vector ^→V is at point P ( 5,-1,1) find the coordinates of the the tip

Let the coordinates of the tip is at point Q( x1,y1,z1) and the vector ^→V = 7i - 6j + k then

x1 - 5 = 7 , x1 = 7+5 = 12, and y1 -(-1) = -6 , y1 +1 = -6, y1 = -6-1 = -7, z1 - 1 = 1, z1 = 1+1 = 2

The coordinates of the tip point Q ( 12, -7, 2)

A quick check p( 5,-1,1) , Q ( 12, -7, 2) , ^→PQ = ( 12-5) i + (-7 -(-1)) j + (2-1)k = 7i -6j + k = ^→V Checks

4- If the tail of ^→V at point P ( x, y, z) and the tip at point Q ( 0,-5,-2)

The coordinates of the tail point are 0 -x = 7, x = -7, and -5 - y = -6, -y = -6+5 = -1, and y =1 , and -2-z = 1, -z = 1+2 , -z = 3, z = -3 , P ( -7 , 1, -3) lets do a quick check

The coordinates of the tail point P( -7,1,-3)

P ( -7, 1, -3) , Q ( 0, -5,-2) , ^→PQ = ( 0 -(-7)) i + ( -5-1) j + ( -2-(-3))k = 7i -6j + k = ^→V , Checks

5- the vector that has the same magnitude V that points in opposite direction is the vector V reversed tip to tail, so if vector ^→AB was vector ^→V the -ve ^→V = ^→BA = (-2-5) i + (4-(-2)) j + (-5-(-4)) k = -7i + 6j - k

or all you have to do is multiply ^→V by -1

6- The vector is the same direction as in ^→V but twice as long is 2(^→V) = 2 ( 7i -6j + k) = 14i - 12j + 2 k