A, B are two non zeros vectors and ||A+b||=||A-B||, then......&nbsp;

The equation given is ||A+B||=||A−B||, where A and B are two non-zero vectors.By squaring both sides of the equation, we get:(||A+B||)2=(||A−B||)2Expanding the squared norms, we have:(A+B)·(A+B)=(A−B)·(A−B)Simplifying the equation, we get:A·A+2A·B+B·B=A·A−2A·B+B·BThe cross terms 2A·B and −2A·B cancel out, resulting in:A·B=−A·BNow, we can add A·B to both sides:A·B+A·B=−A·B+A·B2A·B=0Dividing both sides by 2, we have:A·B=0Therefore, the equation ||A+B||=||A−B|| implies that the dot product of vectors A and B is zero. In other words, A and B are orthogonal or perpendicular to each other.If ||A+B||=||A−B||, then the vectors A and B are orthogonal to each other, and their dot product is zero.

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Maya Maged

Answered question

2022-08-25

A, B are two non zeros vectors and ||A+b||=||A-B||, then......

Answer & Explanation

xleb123

Skilled2023-06-02Added 181 answers

The equation given is

| | A + B | | = | | A - B | |

, where

A

and

B

are two non-zero vectors.
By squaring both sides of the equation, we get:

(| | A + B | |)^{2} = (| | A - B | |)^{2}

Expanding the squared norms, we have:

(A + B) \cdot (A + B) = (A - B) \cdot (A - B)

Simplifying the equation, we get:

A \cdot A + 2 A \cdot B + B \cdot B = A \cdot A - 2 A \cdot B + B \cdot B

The cross terms

2 A \cdot B

and

- 2 A \cdot B

cancel out, resulting in:

A \cdot B = - A \cdot B

Now, we can add

A \cdot B

to both sides:

A \cdot B + A \cdot B = - A \cdot B + A \cdot B

2 A \cdot B = 0

Dividing both sides by 2, we have:

A \cdot B = 0

Therefore, the equation

| | A + B | | = | | A - B | |

implies that the dot product of vectors

A

and

B

is zero. In other words,

A

and

B

are orthogonal or perpendicular to each other.
If

| | A + B | | = | | A - B | |

, then the vectors

A

and

B

are orthogonal to each other, and their dot product is zero.

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