A vector of magnitude 6 and another vector have a resultant of magnitude 12.The vector :
A)must have a magnitude of at least 6 but no more than 18
B)may have a magnitude of 20
C)cannot have a magnitude greater than 12
D)must be perpendicular to
E)must be perpendicular to the resultant vector

Question

A vector  of magnitude 6 and another vector  have a resultant of magnitude 12.The vector  :
A)must have a magnitude of at least 6 but no more than 18 
B)may have a magnitude of 20 
C)cannot have a magnitude greater than 12 
D)must be perpendicular to 
E)must be perpendicular to the resultant vector

Quizplus · Accepted Answer

Let the two given vectors be A and B. Let the unknown vector be C.
Given, ||A||= ||B||= 6 and ||A+B||= 12
Using the triangle inequality, we have ||A+B|| ≤ ||A|| + ||B|| which gives 12 ≤ 12, which is true.
Hence, the three vectors form a triangle and we can use the Law of Cosines to find ||C||.
||C||²= ||A||²+ ||B||² - 2 ||A|| ||B|| cosθ
where θ is the angle between A and B.
cosθ= (||A||²+ ||B||²-||A+B||²) / 2 ||A|| ||B|| = -1/2
||C||²= 2||A||²+ 2||B||² + 2||A|| ||B|| cosθ = 36
Hence, ||C||= 6√2 which satisfies option A.

A Vector of Magnitude 6 and Another Vector