Prove that (\(\vec a+\vec b\)).(\(\vec a+\vec b\))=|\(\vec a\)|2+|\(\vec b\)|2, if and only if \(\vec a\),\(\vec b\) are perpendicular, given a≠0,b≠0.
Prove that (\(\vec a+\vec b\)).(\(\vec a+\vec b\))=|\(\vec a\)|2+|\(\vec b\)|2, if and only if \(\vec a\),\(\vec b\) are perpendicular, given a≠0,b≠0.
Solution and Explanation
(\(\vec a+\vec b\)).(\(\vec a+\vec b\))=\(\vec a\)|2+|\(\vec b\)|2
⇔ \(\vec a.\vec a\)+\(\vec a.\vec b\)+\(\vec b.\vec a\)+\(\vec b.\vec b\)=|\(\vec a\)|2+|\(\vec b\)|2 [Distributivity of scalar products over addition]
⇔|\(\vec a\)|2+2\(\vec a\).\(\vec b\)+|\(\vec b\)|2=|\(\vec a\)|2+|\(\vec b\)|2 [\(\vec a.\vec b=\vec b.\vec a\)(Scalar product is commutative)]
⇔\(2\vec a.\vec b\)=0
⇔\(\vec a.\vec b\)=0
∴\(\vec a\) and \(\vec b\) are perpendicular. [\(\vec a\)≠\(\vec 0\),\(\vec b\)≠\(\vec 0\)(Given)]
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What is the Planning Process?
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Concepts Used:
Vector Algebra
A vector is an object which has both magnitudes and direction. It is usually represented by an arrow which shows the direction(→) and its length shows the magnitude. The arrow which indicates the vector has an arrowhead and its opposite end is the tail. It is denoted as
The magnitude of the vector is represented as |V|. Two vectors are said to be equal if they have equal magnitudes and equal direction.
Vector Algebra Operations:
Arithmetic operations such as addition, subtraction, multiplication on vectors. However, in the case of multiplication, vectors have two terminologies, such as dot product and cross product.