| Example . Find the acute angle between the straight lines and state whether they are parallel or perpendicular. Comparing the given lines with the general Cartesian equations of straight lines, we find ( , ( , , ) b b b ( , , ) d d . Therefore, the acute angle between the two straight lines is θ = cos | ( )( ) ( )( )( ) | + − + − + − = cos ( ) Thus the two straight lines are perpendicular.
Example . Show that the straight line passing through the points ( , , ) ( , , ) B is perpendicular to the straight line passing through the points ( , , ) C ( , , ) D The straight line passing through the points ( , , ) ( , , ) B is parallel to the vector AB OB OA and the straight line passing through the points ( , , ) C ( , , ) D is parallel to the vector CD . Therefore, the angle between the two straight lines is the angle between the two vectors b and d . Since b d ( ) ( ⋅− the two vectors are perpendicular, and hence the two straight lines are perpendicular.
Vector - - Applications of Vector Algebra Aliter We find that direction ratios of the straight line joining the points ( , , ) ( , , ) B are ( , ( , , ) b b b and direction ratios of the line joining the points ( , , ) C ( , , ) D are ( , , ) d d . Since ( )( ) ( )( ) ( )( ) b d b d b d , the two straight lines are perpendicular. Example . Show that the lines are parallel.
We observe that the straight line is parallel to the vector the straight line is parallel to the vector Since ( ) , the two vectors are parallel, and hence the two straight lines are parallel.