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Three-dimensional questions often ask for the foot of the perpendicular from a point to a line, the distance from the point to the line, or the point's mirror image in the line. All three come from one short method built on the dot product. This post sets out the method, works two examples with full checks, and lists the slips that cost marks.
Where is this in the syllabus?
The JEE (Main) 2026 syllabus lists direction ratios, the equation of a line, and skew lines in three dimensional geometry. The JEE (Advanced) 2026 brochure lists lines in space and adds planes. The CBSE 2026-27 curriculum covers the Cartesian and vector equations of a line in Class 12. NCERT's chapter on three dimensional geometry (reprint 2026-27) gives the condition this method rests on: lines with direction ratios a1, b1, c1 and a2, b2, c2 are perpendicular when a1a2 + b1b2 + c1c2 = 0. Check the documents for your exam year.
What is the method?
For a point P and a line through A with direction ratios b = (a, b, c):
General point. Write Q = A + t·b, so each coordinate is linear in t.
Vector PQ. Subtract P from Q.
Perpendicular. Set PQ · b = 0 and solve for t.
Answers. Q is the foot, |PQ| is the distance, and the image is P′ = 2Q − P, because Q is the midpoint of PP′.
Worked example 1: a line in symmetric form
Find the foot, distance and image of P(6, 1, 6) in the line (x − 1)/1 = (y − 2)/2 = (z − 3)/2.
General point: Q = (1 + t, 2 + 2t, 3 + 2t), with b = (1, 2, 2).
PQ = (t − 5, 2t + 1, 2t − 3).
Dot product: 1(t − 5) + 2(2t + 1) + 2(2t − 3) = 9t − 9 = 0, so t = 1.
Foot: Q = (2, 4, 5).
Distance: PQ = (−4, 3, −1), so |PQ| = √(16 + 9 + 1) = √26 ≈ 5.10.
Image: P′ = 2(2, 4, 5) − (6, 1, 6) = (−2, 7, 4).
Checks. PQ · b = −4 + 6 − 2 = 0. The midpoint of P and P′ is ((6 − 2)/2, (1 + 7)/2, (6 + 4)/2) = (2, 4, 5), which is Q. For the distance, use the cross product: with A = (1, 2, 3), AP = (5, −1, 3) and AP × b = (−8, −7, 11). Its length is √(64 + 49 + 121) = √234, and dividing by |b| = 3 gives √234/3 = √26.
Worked example 2: a line through two points
Find the foot and image of P(0, 3, 1) in the line through A(1, 0, 2) and B(3, −1, 3).
Direction: b = B − A = (2, −1, 1).
General point: Q = (1 + 2t, −t, 2 + t).
PQ = (1 + 2t, −t − 3, 1 + t).
Dot product: 2(1 + 2t) − (−t − 3) + (1 + t) = 6t + 6 = 0, so t = −1.
Foot: Q = (−1, 1, 1), and PQ = (−1, −2, 0), so the distance is √5.
Image: P′ = 2(−1, 1, 1) − (0, 3, 1) = (−2, −1, 1).
Check: PQ · b = −2 + 2 + 0 = 0.
Which mistakes cost marks?
Dotting with a point instead of the direction. Use b, never A, in PQ · b = 0.
Misreading the direction ratios. In (1 − x)/2 = y/3 = z/4, the x-term must be rewritten as (x − 1)/(−2), so b = (−2, 3, 4).
Getting the image wrong. It is 2Q − P, not Q − P or P − Q.
Using direction ratios in a distance formula that needs direction cosines. The dot-product method avoids this.
Not checking. If PQ · b is not zero, an arithmetic slip happened earlier.
For the dot and cross products behind every step, see our posts on vector algebra and 3D lines and distances. The image of a point in a plane uses the same midpoint idea with the plane's normal; planes are listed for JEE Advanced, not JEE Main.
A practice routine
Chapter-wise practice hides the choice of method. Rohrer and Taylor (2007) found that students who practised one problem type at a time scored 89% during practice against 60% for mixed practice, but a week later the mixed group scored 63% on the test and the blocked group 20%.
Four-step drill: three questions a session, writing the step names before any numbers.
Two checks per answer: PQ · b = 0 and the midpoint test.
Second route: confirm one distance per week with the cross-product formula.
Mix: put these alongside shortest-distance and angle questions from the same chapter.
If you cannot picture why PQ must be perpendicular to the line, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can revisit the idea behind the dot product.
Today, find the image of the origin in the line (x − 1)/1 = (y − 2)/2 = (z − 3)/2 and check that the midpoint lies on the line.
