Question

Four Vectors (A,B,C,D) all have the same magnitude and lie in a plane. The angle between adjacent vectors is 45 degrees. Which of the following equation is/are correct (A) A-C = -√2 D, (B) B + D - √2C = 0 (C) A+B = B+D (D) A+C/√2 = B​

Answer 1

Answer: Option A and Option B are correct.

Explanation:

Let the magnitude of all vectors be r

A = r i

B = r cos(45°) i + r sin(45°) j = r/√2 i + r/√2 j

C = r j

D = r cos(135°) i + r sin(135°) j =  - r/√2 i + r/√2 j

Option (A) : A - C =√2 D

LHS :       r i - r j

RHS :   -√2(- r/√2 i + r/√2 j ) =  r i - r j

LHS = RHS , it is correct.

Option (B) : B + D -√2 C = 0

LHS :  ( r/√2 i + r/√2 j ) + (- r/√2 i + r/√2 j) - √2r j = 0

RHS :  0

LHS = RHS , it is correct.

Option (C) : A+B = B+D

A ≠ D Therefore, it is incorrect.

Option (D) : A+C/√2 = B​

LHS :  r i +  (r j)/√2  

RHS : r/√2 i + r/√2 j

LHS ≠  RHS , it is incorrect.

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Answer 2

Four Vectors (A,B,C,D) all have the same magnitude and lie in a plane. The angle between adjacent vectors is 45 degrees. Which of the following equation is/are correct (A) A-C = -√2 D, (B) B + D - √2C = 0 (C) A+B = B+D (D) A+C/√2 = B​

Option A AND b are correct.

Solution -

Let's say the length of A is called a, that is, a = | A |.

In rectangular coordinates,

A = (a, 0). Then B = (a/√2, a/√2), C = (0, a), and D = (-a/√2, a/√2).

So then calculate B + D - √2 C

by adding them component-wise B + D - √2 C

= (a/√2, a/√2) + (-a/√2, a/√2) - √2 (0, a)

= (a/√2 - a/√2 - 0, a/√2 + a/√2 - a √2)

= (0, 2a/√2 - a √2 )

= (0, a √2 - a √2)

= (0, 0)

= 0 (the zero vector)

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