Vectors and Three D Geometry Basic Questions for AGNIVEER NAVY

Position Vectors and Direction Cosines

A position vector locates a point in 3D space using coordinates (x, y, z). Direction cosines are the cosines of angles a line makes with the three coordinate axes. They always satisfy l² + m² + n² = 1, where l, m, n are direction cosines. Think of direction cosines as 'direction fingerprints' — every line has a unique set. Direction ratios are proportional to direction cosines but don't have to sum to 1.

• Position vector of point (x, y, z) is r = xi + yj + zk
• Direction cosines l, m, n satisfy l² + m² + n² = 1 always
• Direction ratios a:b:c convert to direction cosines by dividing by sqrt(a² + b² + c²)
• If a line passes through (x₁, y₁, z₁) with direction ratios a:b:c, parametric form is (x₁+at, y₁+bt, z₁+ct)
• Magnitude of vector v = sqrt(x² + y² + z²)

Dot Product (Scalar Product)

Dot product of two vectors gives a scalar (single number). It measures how much two vectors point in the same direction. If vectors are perpendicular, dot product is zero — this is the key test for perpendicularity. Dot product is commutative: a·b = b·a. Geometrically, a·b = |a||b|cos(θ), where θ is angle between them.

• a·b = a₁b₁ + a₂b₂ + a₃b₃ (component form)
• a·b = |a||b|cos(θ) (geometric form)
• If a·b = 0, vectors are perpendicular
• Dot product is always a number, never a vector
• Used to find angle between lines, check perpendicularity, find projections

Cross Product (Vector Product)

Cross product of two vectors gives another vector perpendicular to both. It's anti-commutative: a × b = -(b × a). The magnitude |a × b| = |a||b|sin(θ) equals the area of parallelogram formed by the two vectors. Cross product is zero if vectors are parallel. Use determinant method with i, j, k for calculation.

• a × b is perpendicular to both a and b
• a × b = -(b × a) (order matters — anti-commutative)
• If a × b = 0, vectors are parallel
• |a × b| = |a||b|sin(θ) = area of parallelogram
• Calculated using 3×3 determinant with i, j, k as first row

Distance Between Points and Lines

Distance between two points is straightforward: use the distance formula. Distance from a point to a line requires the perpendicular from the point to the line. Distance between two skew lines (non-parallel, non-intersecting lines in 3D) uses a special formula involving cross product. These are high-frequency exam questions.

• Distance between P(x₁,y₁,z₁) and Q(x₂,y₂,z₂): sqrt((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²)
• Distance from point to line uses perpendicular foot concept
• Distance between skew lines: |((r₂-r₁)·(d₁×d₂))| / |d₁×d₂|, where r₁, r₂ are position vectors and d₁, d₂ are direction vectors
• If lines intersect, distance = 0
• If lines are parallel, use point-to-line formula

Equation of Line and Plane

A line in 3D can be written in vector form r = a + λb (where a is position vector of a point, b is direction vector, λ is parameter) or Cartesian form (x-x₁)/l = (y-y₁)/m = (z-z₁)/n. A plane is defined by a normal vector and a point, or by equation ax + by + cz + d = 0. Exam questions often ask for equation of line through two points or plane through three points.

• Vector form of line: r = a + λb
• Cartesian form of line: (x-x₁)/l = (y-y₁)/m = (z-z₁)/n
• Equation of plane: a(x-x₁) + b(y-y₁) + c(z-z₁) = 0, where (a,b,c) is normal vector
• General plane equation: ax + by + cz + d = 0
• To find line through two points, use their coordinates to get direction ratios

• Confusing direction cosines with direction ratios — direction cosines ALWAYS satisfy l²+m²+n²=1, direction ratios do not. Always normalize direction ratios first.
• Getting cross product sign wrong — a × b ≠ b × a. The order matters. Many students forget the negative sign in anti-commutativity.
• Using dot product formula when cross product is needed, or vice versa — dot product gives a scalar (check perpendicularity), cross product gives a vector (check parallelism or find area).
• Forgetting to take magnitude in distance formulas — distance is always positive. Don't forget the square root or absolute value bars.
• Mixing up parametric and Cartesian forms — parametric uses parameter t or λ, Cartesian uses equal ratios. Know which form the question asks for.

• DOT = scalar, CROSS = vector. Dot product gives a number (perpendicular when zero). Cross product gives a vector (parallel when zero).
• Direction Cosines: l² + m² + n² = 1 ALWAYS. Think 'Pythagorean constraint' — they're like coordinates on a unit sphere.
• Cross Product Determinant: First row is i, j, k. Second row is first vector components. Third row is second vector components. Expand like a 3×3 matrix.
• Distance Between Skew Lines: Remember the formula has cross product in numerator (magnitude) and denominator (normalizer). It's the 'perpendicular sandwich' formula.

• Agniveer Navy typically asks 2-3 questions on vectors. Expect one straightforward dot/cross product calculation, one distance problem, and possibly one direction cosines question.
• Time pressure is real — these questions should take 2-3 minutes max. Memorize the determinant expansion for cross product; don't derive it during exam.
• Recent papers show preference for numerical answers (e.g., 'distance is sqrt(89)') over symbolic. Always simplify your final answer.
• Direction cosines questions often appear as 'find direction cosines of line through two points' — convert to direction ratios first, then normalize.
• Skew lines and distance between lines are less common but high-value if you know the formula. If you see it, you can solve it quickly and gain an edge.