If two vectors \(\vec{A}=6\hat{i}-8\hat{j}+4\hat{k}\) and \(\vec{B}=4\hat{i}-6\hat{j}+p\hat{k}\) are mutually perpendicular, then value of p is

CONCEPT:

The dot product of vector:

• The dot product is the sum of the products of the corresponding entries of the two sequences of numbers.
• Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.

\({{\rm{A}}_1}\cdot{{\rm{A}}_2} = \left| {\overrightarrow {{{\rm{A}}_1}} } \right|\left| {\overrightarrow {{{\rm{A}}_2}} } \right|\cos {\rm{θ }}\)

Where \(\left| {\overrightarrow {{{\rm{A}}_1}} } \right|.\left| {\overrightarrow {{{\rm{A}}_2}} } \right|\) are the magnitudes of two vectors A1 and A2

CALCULATION:

Given - \(\vec{A}=6\hat{i}-8\hat{j}+4\hat{k}\) ,  \(\vec{B}=4\hat{i}-6\hat{j}+p\hat{k}\) and θ - 90° 

• Here the two vectors are perpendicular, therefore the scalar product is equal to zero  i.e., 

\(⇒ \left| {\overrightarrow {{{\rm{A}}_1}} } \right|\left| {\overrightarrow {{{\rm{A}}_2}} } \right|\cos {\rm{90^\circ =0}}\)

• The dot product of vector is 

\(⇒ (6\hat{i}-8\hat{j}+4\hat{k})\cdot(4\hat{i}-6\hat{j}+p\hat{k})=0\)

\(⇒ 24+48+4p = 0\)

⇒ p = -18

Parallel Vectors:

• “Two vectors \(\vec A\) and \(\vec B\) are parallel if and only if they are scalar multiples of one another.”

OR 

• If the angle between two vectors is 0° or 180°, then these two vectors are said to be parallel.