Vector Algebra

Vectors have both magnitude and direction, unlike scalar which has only magnitude. Real life full of vectors: Navigation - aeroplane flying from Delhi to Mumbai direction matters not just distance, pilot uses vector displacement. Physics force is vector - you push a box force direction matters. Wind velocity is vector - affects flight path. Engineering bridge load forces are vectors. Gaming - character moves in game world using position vector, velocity vector. GPS - your movement is vector. Mobile phone screen touch drag is vector. Sports cricket ball trajectory is vector. So vector algebra helps us add forces, find work, find area, find direction. That is why vectors powerful.

Vector is quantity with magnitude and direction, represented by arrow from initial point to terminal point, denoted as AB with arrow or a with arrow. Magnitude is length of arrow |a|. Types: Zero vector magnitude 0 direction arbitrary, denoted 0. Unit vector magnitude 1 in direction of a, denoted a cap = a/|a|. Equal vectors same magnitude same direction. Negative vector opposite direction same magnitude -a. Collinear vectors parallel same or opposite direction, lying on same line or parallel lines. Coplanar vectors lie in same plane. Position vector of point P with coordinates (x,y,z) is OP = x i + y j + z k from origin O to P. i,j,k are unit vectors along x,y,z axes.

Addition of vectors means combining two vectors to get resultant vector that has same effect. Triangle law: Place second vector b at head of first vector a, resultant R = a + b connects tail of a to head of b, forms triangle. Parallelogram law: Place a and b with same tail origin, complete parallelogram, diagonal from common origin is resultant R = a + b. Properties: Commutative a + b = b + a, Associative (a + b) + c = a + (b + c), Zero vector a + 0 = a, Additive inverse a + (-a) = 0, Triangle inequality |a + b| ≤ |a| + |b| magnitude of sum less than sum of magnitudes unless same direction.

Components: Any vector v in xy plane can be written as v = x i + y j where x is x-component along i, y is y-component along j. In 3D v = x i + y j + z k. Multiplication by scalar k: Multiply each component by k, k v = (k x, k y, k z). If k>0 direction same, k<0 direction opposite, magnitude multiplied by |k|. Example v=(3,4) k=2 k v=(6,8). Position vector of point A with position vector a and B with b, point P dividing AB internally in ratio m:n has position vector r = (m b + n a)/(m+n). If m=n=1 P is midpoint r=(a+b)/2. Externally r=(m b - n a)/(m-n). This section formula helps find coordinates of point dividing line.

Dot product of two vectors a and b is scalar (number) defined as a·b = |a||b| cos θ where θ is angle between a and b (0≤θ≤π). If θ=0 cos0=1 dot max |a||b|, if θ=90° cos90=0 dot 0 perpendicular. Dot product gives projection and work. Formula in components if a=a1 i+a2 j+a3 k and b=b1 i+b2 j+b3 k then a·b = a1 b1 + a2 b2 + a3 b3. Properties: Commutative a·b=b·a, Distributive a·(b+c)=a·b+a·c, a·a=|a|², if a·b=0 then a⊥b perpendicular (unless zero). Angle cos θ = (a·b)/(|a||b|). Projection of a on b = (a·b)/|b| times unit vector b cap, scalar projection = a·b/|b|. Work done W = Force·displacement = F·d.

Cross product of two vectors a and b is vector defined as a×b = |a||b| sin θ n cap where θ is angle between a and b, n cap is unit vector perpendicular to plane containing a and b following right hand rule (fingers a to b thumb n). Magnitude |a×b|=|a||b| sin θ = area of parallelogram with sides a,b. Direction perpendicular to both a and b. Formula components using determinant: a×b = |i j k; a1 a2 a3; b1 b2 b3| = (a2 b3-a3 b2)i - (a1 b3-a3 b1)j + (a1 b2-a2 b1)k. Properties: Not commutative a×b = - b×a opposite direction, Distributive, a×a=0 zero vector, if a×b=0 then a parallel to b (θ=0 or 180). Torque τ = r×F, Force producing rotation.

Scalar triple product of three vectors a,b,c is defined as [a b c] = a·(b×c) = (a×b)·c which gives scalar. Geometrically it is volume of parallelepiped with coterminous edges a,b,c as sides. Formula using determinant: If a=(a1,a2,a3) b=(b1,b2,b3) c=(c1,c2,c3) then [a b c]= determinant |a1 a2 a3; b1 b2 b3; c1 c2 c3|. Volume of parallelepiped = |[a b c]| absolute value. Volume of tetrahedron with same edges = 1/6 |[a b c]|. Properties: Cyclic [a b c]=[b c a]=[c a b], if any two vectors equal triple product 0, if a,b,c coplanar volume 0 so [a b c]=0 condition for coplanarity. If [a b c]=0 then a,b,c coplanar.

Summary: Scalar only magnitude mass, time, speed. Vector magnitude+direction displacement, force, velocity. Types zero, unit, equal, negative, collinear, coplanar. Addition triangle law b at head of a resultant tail to head, parallelogram diagonal. Properties commutative, associative, triangle inequality. Components v=x i+y j+z k. Scalar multiplication k v scales magnitude. Position vector OP = x i+y j+z k. Section formula internal r=(m b+n a)/(m+n) midpoint (a+b)/2. Dot product a·b=|a||b| cos θ scalar a1 b1+a2 b2+a3 b3 angle cos θ=(a·b)/(|a||b|) perpendicular if 0 work F·d projection. Cross product a×b=|a||b| sin θ n cap vector area parallelogram |a×b| anti-commutative -b×a torque r×F. Triple product [a b c]=a·(b×c)=det volume parallelepiped |[a b c]| tetrahedron 1/6. Choose dot for angle/work/projection, cross for area/torque/perpendicular vector.