Given: point P(1, 2, 3)
To find: coordinates of the foot of the perpendicular from the point on the y-axis
Formula used:
The distance between any two points (a, b, c) and (m, n, o) is given by,
\(\sqrt{(a-m)^2+(b-n)^2+(c-o)^2}\)
As x and z coordinate on the y-axis is zero
Let point D is the point of the foot of perpendicular on the y-axis from point P be (0, y, 0)
Direction cosines of y-axis are (0, 1, 0)
Direction cosines of PD are (1 – 0, 2 – y, 3 – 0) = (1, 2 – y, 3)
Let \(\overrightarrow{b_1}\) are \(\overrightarrow{b_2}\) two vectors as shown in the figure:
The dot product of perpendicular vectors is always zero
Therefore, \(\overrightarrow{b_1}. \overrightarrow{b_2}\) = 0
⇒ 1 × 0 + (2 – y) × 1 + 3 × 0 = 0
⇒ 0 + 0 + 2 – y= 0
⇒ y = 2
Hence, coordinates of point D are (0, 2, 0)
