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Mathematics · Class 11
Add vectors componentwise (or tip-to-tail / parallelogram). Triangle law: a + b is the third side. Scalar multiply: stretch or shrink, reverse if the scalar is negative. Section formula / midpoint: midpoint of AB is a+b2 if a,b are position vectors. Meaning: addition is 'follow one arrow, then the other'.
PracticeAdd (or subtract) entrywise, only if the sizes match. Scalar k times A multiplies every entry by k. Negative: -A = (-1)A. Meaning: addition is 'same box, add matching numbers'. It is not matrix multiplication (next page).
PracticeAllied angles are built from A by adding or subtracting 90, 180, 270, 360 (and negatives). Examples: sin(180-A)=sin A, cos(180-A)=-cos A, tan(180-A)=-tan A. sin(90-A)=cos A. The CAST (or ASTC) rule names which functions are positive in which quadrant. Meaning: reduce to an acute related angle, then attach the right sign.
PracticePlot z = x + i y as the point (x, y). The modulus r = |z| is the distance from the origin. The argument theta is the angle from the positive real axis to the ray OP, usually in radians. Polar form: z = r (cos theta + i sin theta). Euler: e^{i theta} = cos theta + i sin theta, so z = r e^{i theta}. Meaning: same point, described by length and turn instead of x and y.