Question

What is the numerical value of vector \[3\overset{\hat{\ }}{\mathop{i}}\,+4\overset{\hat{\ }}{\mathop{j}}\,+5\overset{\hat{\ }}{\mathop{k}}\,\]?

Answer 1

Hint: We are given with a vector. So in order to find the numerical value of the vector given i.e. \[3\overset{\hat{\ }}{\mathop{i}}\,+4\overset{\hat{\ }}{\mathop{j}}\,+5\overset{\hat{\ }}{\mathop{k}}\,\], we must apply the formula of vectors which helps us in computing the value of the vectors into numerical form. We can find it using the formula \[\overset{-}{\mathop{A}}\,=a\overset{\hat{\ }}{\mathop{i}}\,+b\overset{\hat{\ }}{\mathop{j}}\,+c\overset{\hat{\ }}{\mathop{k}}\,\Rightarrow \left| \overset{-}{\mathop{A}}\, \right|=\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}\]. Upon substituting and solving, we obtain the required solution.

Complete step by step answer:
Now let us have a brief regarding the vectors. A vector is nothing but an object that has both magnitude and direction. In the symbolic representation of a vector, the length represents the magnitude and the arrow indicates the direction of the vector. When we add the vector, we obtain the same result irrespective of the order we add them. We can also perform the subtraction upon the vectors. When the vectors are broken up, they are termed as components. The magnitude of a vector is denoted by two bars on either side of the vector.
Now let us find out the numerical value of the given vector \[3\overset{\hat{\ }}{\mathop{i}}\,+4\overset{\hat{\ }}{\mathop{j}}\,+5\overset{\hat{\ }}{\mathop{k}}\,\]
Let us apply the vector formula.
\[\overset{-}{\mathop{A}}\,=a\overset{\hat{\ }}{\mathop{i}}\,+b\overset{\hat{\ }}{\mathop{j}}\,+c\overset{\hat{\ }}{\mathop{k}}\,\Rightarrow \left| \overset{-}{\mathop{A}}\, \right|=\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}\]
We can compare our vector with the general form of the vector.
\[a\overset{\hat{\ }}{\mathop{i}}\,+b\overset{\hat{\ }}{\mathop{j}}\,+c\overset{\hat{\ }}{\mathop{k}}\,\]\[=3\overset{\hat{\ }}{\mathop{i}}\,+4\overset{\hat{\ }}{\mathop{j}}\,+5\overset{\hat{\ }}{\mathop{k}}\,\]
Upon substituting, we get
\[\left| \overset{-}{\mathop{A}}\, \right|=\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}\]
\[\Rightarrow \sqrt{{{3}^{2}}+{{4}^{2}}+{{5}^{2}}}=\sqrt{9+16+25}=\sqrt{50}\]
\[\therefore \] The numerical value of the vector \[3\overset{\hat{\ }}{\mathop{i}}\,+4\overset{\hat{\ }}{\mathop{j}}\,+5\overset{\hat{\ }}{\mathop{k}}\,\]\[=\sqrt{50}\]

Note: We must check if the vector we have is the same as the vector in the general form. If any of the components is absent, then we must consider it as zero and then calculate. This consideration does not bring out any difference while adding the vectors, but while subtracting there might occur a change in the direction of the vectors.