3D Geometry
Perpendicular from Point to Line
Grade 12
Question:
<p>The coordinates of the foot of the perpendicular drawn from the point \(A(1, 0, 3)\) to the join of the points \(B(4, 7, 1)\) and \(C(3, 5, 3)\) are</p>
<p>(a) \(\left(\frac{5}{3}, \frac{7}{3}, \frac{17}{3}\right)\)</p>
<p>(b) \((5, 7, 17)\)</p>
<p>(c) \(\left(\frac{5}{7}, -\frac{3}{7}, \frac{17}{3}\right)\)</p>
<p>(d) \(\left(-\frac{5}{3}, \frac{7}{3}, -\frac{17}{3}\right)\)</p>
Step-by-Step Solution
Key Concept: Use the condition that the foot of perpendicular divides the line segment such that the vector from the external point to the foot is perpendicular to the line.
Step 1: Let \(D\) be the foot of the perpendicular and let it divide \(BC\) in the ratio \(λ:1\). The coordinates of \(D\) are: \[\left(\frac{3λ + 4}{λ + 1}, \frac{5λ + 7}{λ + 1}, \frac{3λ + 1}{λ + 1}\right)\] Step 2: For \(D\) to be the foot of perpendicular, \(\vec{AD} ⊥ \vec{BC}\) \(\vec{AD} · \vec{BC} = 0\) Step 3: This gives us: \(-(2λ + 3) - 2(5λ + 7) - 4 = 0\) Step 4: Solving: \(λ = -\frac{7}{4}\) Step 5: Substituting back, the coordinates of \(D\) are \(\left(\frac{5}{3}, \frac{7}{3}, \frac{17}{3}\right)\) Answer is (a).
Correct Answer: A