Question

Show that the sum of the squares of the direction cosines of a line through the origin is equal to 1 Hint: Let $\left(a,b,c\right)$be a point on the line at distance 1 from the origin. Write the direction cosines in terms of $\left(a,b,c\right)$.

Short answer

Answer

The statement has been verified.

Step-by-step solution

Given Information

The line passes through the origin.

Definition of Direction cosine.

The direction cosines of a vector are the cosines of the angles between the vector and the three axes.

Prove the statement.

The line passes through the origin.

Letbe the vector of unit length.

$r=ai+bj+ck$

Find the direction cosines.

$$\begin{gathered} r.i-a \\ r.j=b \\ r.k=c \end{gathered}$$

Since r and base vectors both are unit vectors.

$$\begin{gathered} a^2+b^2+c^2={\left|r\right|}^2 \\ {a}^2+b^2+c^2=1 \end{gathered}$$

Hence, the statement has been verified.