Example 26 - Chapter 10 Class 12 Vector Algebra

Transcript

Example 26 Write all the unit vectors in XY-plane Let the unit vector be We know that = + y + z Since the vector is in X-Y plane, there is no Z coordinate. Hence, = x + y + 0 = + y Taking a general vector , making an angle with the x axis Unit vector in direction of x axis is & in y axis is Given that makes an angle of with x-axis So, angle between & is Now, we know that, . = cos , Putting = , = & = . = cos . = 1 1 cos . = cos ( + y + 0 ). = cos ( + y + 0 ). (1 + 0 + 0 ) = cos .1 + y.0 + 0.0 = cos x = cos Similarly, makes an angle of (90 ) with y-axis So, angle between & is (90 ) Now, we know that, . = cos , Putting = , = & = (90 ) . = cos (90 ) . = 1 1 cos (90 ) . = cos (90 ) . = sin ( + y + 0 ). = sin ( + y + 0 ). (0 + 1 + 0 ) = sin .0 + y.1 + 0.0 = sin y = sin So, x = cos and y = sin Thus, = x + y = cos + sin = cos + sin This value will be true in all quadrants Quadrant 1: 0 < 90 Quadrant 2: 90 < 180 Quadrant 3: 180 < 270 Quadrant 4: 270 < 360 So, 0 360 i.e. 0 2 So, = cos + sin ; for 0 2