86. Let a and b be two non-zero vectors such that
|a + b| = |a − b|
Then
(A) a and b are parallel to each other
(B) a and b are perpendicular to each other
(C) a is NOT a scalar multiple of b
(D) a × b = 0
Condition for |a + b| = |a − b| – Complete Concept and Detailed Solution
The present question involves the equality of the magnitudes of two vectors, namely |a + b| and |a − b|. Although this appears to be a simple algebraic identity, it actually reveals an important geometric relationship between the vectors. The solution requires expanding the squared magnitudes using the dot product and interpreting the resulting condition.
Correct Answer
Option (B): a and b are perpendicular to each other.
Understanding the Concept
The magnitude of a vector can be expressed in terms of the dot product. Whenever an equation involves magnitudes of vectors, the most effective strategy is to square both sides. This eliminates the square roots and converts the problem into an algebraic equation involving dot products.
Recall the following identities:
|a+b|²=(a+b)·(a+b)
|a−b|²=(a−b)·(a−b)
Expanding these expressions allows us to compare both sides directly.
Step 1: Square Both Sides
The given condition is
|a+b|=|a−b|.
Squaring both sides gives
|a+b|²=|a−b|².
This step is valid because magnitudes are always non-negative.
Step 2: Expand Both Sides
Using the dot product identities,
|a+b|²=a·a+2a·b+b·b
Therefore,
|a+b|²=|a|²+2(a·b)+|b|².
Similarly,
|a−b|²=|a|²−2(a·b)+|b|².
Step 3: Compare the Two Expressions
Since both squared magnitudes are equal,
|a|²+2(a·b)+|b|²
=
|a|²−2(a·b)+|b|².
Subtracting the common terms from both sides gives
4(a·b)=0.
Hence,
a·b=0.
Step 4: Interpret the Result
The dot product of two non-zero vectors satisfies
a·b=|a||b|cosθ.
Since both vectors are non-zero,
|a|≠0
and
|b|≠0.
Therefore,
cosθ=0.
This happens only when
θ=90°.
Hence, the vectors are perpendicular (orthogonal).
Geometrical Interpretation
The vectors a+b and a−b represent the diagonals of the parallelogram formed by vectors a and b. A well-known geometric property states that the diagonals of a parallelogram are equal in length if and only if the parallelogram is a rectangle.
A rectangle has adjacent sides meeting at right angles. Therefore, the equality
|a+b|=|a−b|
implies that the vectors are perpendicular.
Explanation of Every Option
Option (A): a and b are parallel to each other
This option is incorrect. If two non-zero vectors are parallel, their dot product is generally not zero unless one vector is the zero vector, which is excluded in the question.
Option (B): a and b are perpendicular to each other
This option is correct because the equality of the magnitudes leads directly to a·b=0, which is the defining condition for orthogonality.
Option (C): a is NOT a scalar multiple of b
Although perpendicular non-zero vectors are indeed not scalar multiples of one another, this statement is only a consequence of the result and is not the defining property established by the given condition. The direct conclusion is that the vectors are perpendicular.
Option (D): a × b = 0
This option is incorrect because the cross product of two vectors is zero only when the vectors are parallel or antiparallel. Here, the vectors are perpendicular, so the magnitude of the cross product is actually maximum.
Mathematical Verification
Let
a=(1,0)
b=(0,1).
These vectors are perpendicular.
|a+b|=|(1,1)|=√2.
|a−b|=|(1,−1)|=√2.
Both magnitudes are equal, confirming the derived result.
Alternative Method
Instead of expanding the dot products immediately, one may remember the standard vector identity
|a+b|²−|a−b|²=4(a·b).
Since the two magnitudes are equal, their squares are equal.
Therefore,
4(a·b)=0
which directly gives
a·b=0.
This shortcut is particularly useful during objective examinations.
Related Practice Example
If two non-zero vectors satisfy
|a+b|²=|a|²+|b|²,
then expanding the left-hand side gives
|a|²+2(a·b)+|b|²=|a|²+|b|².
Hence,
a·b=0.
Therefore, the vectors are perpendicular.
This example reinforces the same concept used in the present problem.
Key Takeaways
Whenever an equation involves vector magnitudes, first consider squaring both sides. This transforms the problem into an algebraic equation involving dot products, which can then be interpreted geometrically. Remember that the condition a·b=0 is the defining criterion for perpendicular vectors, while a×b=0 indicates parallel vectors.
Final Answer
Expanding both squared magnitudes gives
a·b=0.
Since the vectors are non-zero, this implies that the angle between them is 90°.
Therefore, the vectors are perpendicular.
Correct Option: (B)
