Why it is said that if cross product of two vectors are zero, the two vectors are parallel and if dot product is zero, the two vectors are perpendicular?

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Solution

Dear Student ,
Let two vectors are $\vec{A} a n d \vec{B} .$
Now the dot product of two vector is ,
$\vec{A} \cdot \vec{B} = 0 \Rightarrow |\vec{A}| |\vec{B}| \cos θ = 0 \Rightarrow \cos θ = 0 = \cos 90 ° \Rightarrow θ = 90 ° S o t h e \vec{A} a n d \vec{B} a r e p e r p e n d i c u l a r t o e a c h o t h e r . A g a i n, i n c a s e o f c r o s s p r o d u c t, \vec{A} \times \vec{B} = 0 \Rightarrow |\vec{A}| |\vec{B}| \sin θ = 0 \Rightarrow \sin θ = 0 = \sin 0 ° \Rightarrow θ = 0 ° T h e r e f o r e t h e \vec{A} a n d \vec{B} a r e p a r a l l e l t o e a c h o t h e r .$
Hope now you understand this .
Regards

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(^i +^j +^k)

(^i - 2^j +^k)

A

\to a \times \to b = \to c \times \to d

\to a \times \to c = \to b \times \to d

\to b - \to c

\to a - \to d

R

If the dot product of two non-zero vectors is zero, then the vectors

\to A a n d \to B

Assertion : If dot product and cross product of vector A and vector B are zero, it implies that one of the vector A and B must be a null vector.

Reason : Null vector is a vector with zero magnitude.

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