Question

How do you find the scalar and vector projections of \[b\] onto \[a\]? Given \[a = i + j + k\], \[b = i - j + k\].

Answer 1

Hint:In the given question, we have been given two vectors. We have to find the scalar and vector projections of one vector onto the other vector. To do that, we apply the formulae of the projections – scalar projection of \[\overrightarrow x \] on \[\overrightarrow y \] means the magnitude of resolved component of \[\overrightarrow x \] in the direction of \[\overrightarrow y \], while vector projection of \[\overrightarrow x \] on \[\overrightarrow y \] means the resolved component of \[\overrightarrow x \] in the direction of \[\overrightarrow y \].

Formula Used:
Scalar projection of \[\overrightarrow x \] on \[\overrightarrow y \]\[ = \dfrac{{\overrightarrow x .\overrightarrow y }}{{\left| {\overrightarrow y } \right|}}\].
Vector projection of \[\overrightarrow x \] on \[\overrightarrow y \]\[ = \dfrac{{\overrightarrow x .\overrightarrow y }}{{{{\left| {\overrightarrow y } \right|}^2}}}.\overrightarrow y \].

Complete step by step answer:
The given vectors are \[\overrightarrow a = i + j + k\] and \[\overrightarrow b = i - j + k\].
Now, the scalar projection of \[\overrightarrow b \] onto \[\overrightarrow a \]\[ = \dfrac{{\overrightarrow a .\overrightarrow b }}{{\left| {\overrightarrow a } \right|}}\]
\[ = \dfrac{{\left( {i + j + k} \right).\left( {i - j + k} \right)}}{{\sqrt {{1^2} + {1^2} + {1^2}} }} = \dfrac{{{1^2} - {1^2} + {1^2}}}{{\sqrt 3 }} = \dfrac{1}{{\sqrt 3 }}\].
Vector projection \[ = \dfrac{{\overrightarrow a .\overrightarrow b }}{{{{\left| {\overrightarrow a } \right|}^2}}}.\overrightarrow a \].
\[ = \dfrac{{\left( {i + j + k} \right).\left( {i - j + k} \right)}}{{\sqrt {{1^2} + {1^2} + {1^2}} }}.\left( {i + j + k} \right) = \dfrac{1}{{\sqrt 3 }}\left( {i + j + k} \right)\].

Note: In the given question, we had to find the scalar and vector projections of one vector onto the other vector. We had been given the values of the vectors. We just simply wrote down the formulae of the two projections, put in the values of the vectors, calculated the result and we got our answer. So, it is really important that we know the formulae and where, when, and how to use them so that we can get the correct result.