Maths · Equation of a line; Skew lines, the shortest distance between them and its equation
The foot of the perpendicular from the point \) to the plane is?
The foot of the perpendicular from the point \( \boldsymbol{A}(\mathbf{7}, \mathbf{1 4}, \mathbf{5}) \) to the plane \( \mathbf{2} \boldsymbol{x}+\mathbf{4} \boldsymbol{y}- \) \( z=2 \) is?
- A. (3,1,8)
- B. (1,2,8)
- C. (3,-3,5)
- D. (5,-3,-4)
Step-by-step solution
The foot of the perpendicular from a point (x1,y1,z1) to a plane ax+by+cz+d=0 is given by (x1 - a*t, y1 - b*t, z1 - c*t) where t = (ax1+by1+cz1+d)/(a^2+b^2+c^2). For plane 2x+4y-z=2, rewrite as 2x+4y-z-2=0, so a=2, b=4, c=-1, d=-2. Point (7,14,5). Compute numerator: 2*7+4*14+(-1)*5 + (-2) = 14+56-5-2=63, denominator: 4+16+1=21, t=3. Then foot: x=7-2*3=1, y=14-4*3=2, z=5-(-1)*3=8. Thus foot is (1,2,8).
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