If a vector 2 i + 3 j + 8 k is perpendicular to the vector 4 j − 4 i + α k then the value of α is? solve without using the property of dot product.

Question

If a vector   2  i  +  3  j  +  8  k is perpendicular to the vector   4  j  −  4  i  +      α    k then the value of α is? solve without using the property of dot product.

periasemdy · Accepted Answer

Since the two vectors are perpendicular, a less natural way would be think of Pythagoras theoremLet                     A        B            →        =  2  i  +  3  j  +  8  k and                     A        C            →        =  −  4  i  +  4  j  +  α  kSince                     A        B            →        ⊥                    A        C            →      , Therefore we have                     B        C            →        =                    A        C            →        −                    A        B            →        =  −  6  i  +  j  +  (  α  −  8  )  kAlso   B      C    2    =  A      C    2    +  A      B    2    (  −  6      )    2    +  (  1      )    2    +  (  α  −  8      )    2    =  (  16  +  16  +      α    2    )  +  (  4  +  9  +  64  )  α  =  −  0.5

4enevi · Accepted Answer

You can use the Pythagorean theorem. If you add two vectors, you get  6  i  −  j  +  (  8  +  α  )  k. Then we have  36  +  1  +  (  8  +  α      )    2    =  4  +  9  +  64  +  16  +  16  +      α    2      ⟹    16  α  =  8

If a vector 2i+3j+8k is perpendicular to the vector 4j−4i+alpha k then the value of alpha is? solve without using the property of dot product.

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