Show that the direction cosines of a vector equally inclined to the axes OX, OY, and OZ are \(\frac{1}{\sqrt 3}\),\(\frac{1}{\sqrt 3}\),\(\frac{1}{\sqrt 3}\).
Show that the direction cosines of a vector equally inclined to the axes OX, OY, and OZ are \(\frac{1}{\sqrt 3}\),\(\frac{1}{\sqrt 3}\),\(\frac{1}{\sqrt 3}\).
Solution and Explanation
Let a vector be equally inclined to axes OX,OY,and OZ at angle a.
Then, the direction cosines of the vector are cos a, cos a, cos a.
Now,
cos2a+cos2a+cos2a=1
\(\Rightarrow\)3cos2a=1
\(\Rightarrow\)cos a=\(\frac{1}{\sqrt 3}\)
Hence, the direction cosines of the vector which are equally inclined to the axes are \(\frac{1}{\sqrt 3}\),\(\frac{1}{\sqrt 3}\),\(\frac{1}{\sqrt 3}\).
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Concepts Used:
Vector Algebra
A vector is an object which has both magnitudes and direction. It is usually represented by an arrow which shows the direction(→) and its length shows the magnitude. The arrow which indicates the vector has an arrowhead and its opposite end is the tail. It is denoted as
The magnitude of the vector is represented as |V|. Two vectors are said to be equal if they have equal magnitudes and equal direction.
Vector Algebra Operations:
Arithmetic operations such as addition, subtraction, multiplication on vectors. However, in the case of multiplication, vectors have two terminologies, such as dot product and cross product.