Dot Product — Angle between vectors, projection
Vectors
12
JEE Qs
8%
Hard
60
min
Master both the algebraic (component form) and geometric (magnitude-angle form) definitions of the dot product to efficiently solve a wide range of problems.
🧮 Key Formulas
✅ Key Points for JEE
- 1The dot product (scalar product) yields a scalar value, representing the geometric 'effect' or component of one vector along the direction of another.
- 2Use the component form `a.b = a_x b_x + a_y b_y + a_z b_z` for algebraic computations and `a.b = |a||b|cos(theta)` for problems involving angles or magnitudes.
- 3Two non-zero vectors are orthogonal (perpendicular) if and only if their dot product is zero; this is a frequent condition in JEE problems.
- 4Distinguish clearly between the scalar projection (a scalar magnitude) and the vector projection (a vector in the direction of the target vector).
- 5The dot product is commutative (a.b = b.a) and distributive over vector addition (a.(b+c) = a.b + a.c), which simplifies complex vector expressions.
⚠️ Common Mistakes
- ✕Confusing scalar projection `(a.b)/|b|` with vector projection `((a.b)/|b|^2) * b`, which are dimensionally different (scalar vs. vector).
- ✕Incorrectly calculating the magnitude of a vector, leading to errors in angle or projection calculations.
- ✕Assuming `a.b = 0` implies orthogonality without considering the case where `a` or `b` might be the null vector.
- ✕Making sign errors or misinterpreting the angle when `cos(theta)` is negative, indicating an obtuse angle.
📝 Practice Questions
See allQ10.Let →a and →b be two unit vectors such that the angle between them is . If λ→a + 2→b and 3→a −λ→b are 3 perpendicular to each other, then the number of values of λ in [−1, 3] is : (1) 2 (2) 1 (3) 0 (4) 3 e 1 x x loge α
Q10.Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC , divides the arc −−−length of arc AB 1 → → → AC such that length of arc BC = 5 , and OC = αOA + βOB , then α + √2(√3 −1)β is equal to (1) 2√3 (2) 2 −√3 (3) 5√3 (4) 2 + √3 f ∘g is
Q23.Let →c be the projection vector of →b = λ^i + 4^k, λ > 0, on the vector →a = ^i + 2^j + 2^k. If |→a + →c| = 7, then the area of the parallelogram formed by the vectors →b and →c is ________
Q4. Let a line pass through two distinct points P(−2, −1, 3) and Q , and be parallel to the vector 3^i + 2^j + 2^k. If the distance of the point Q from the point R(1, 3, 3) is 5 , then the square of the area of △PQR is equal to : (1) 148 (2) 136 (3) 144 (4) 140
Q12.Let →a = 3^i −^j + 2^k, b =→a× (^i −2^k) and→c= b × ^k. Then the projection of→c−2^j on →a is : (1) 2√14 (2) √14 (3) 3√7 (4) 2√7
Q48.Two particles are located at equal distance from origin. The position vectors of those are represented by ¯A = 2^i + 3n^j + 2^k and ¯B = 2^i −2^j + 4p^k, respectively. If both the vectors are at right angle to each other, the value of n−1 is _____ .
NCERT Chapters
- Class 12 Maths Ch 10: Vector Algebra