Mathematical Methods — Flashcards

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Scalars are physical quantities that can be completely described by magnitude alone (e.g., mass, time, temperature). Vectors are physical quantities that need both magnitude and direction for complete…

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A unit vector is a vector having unit magnitude in a given direction. If M is a non-zero vector with magnitude M = |M|, then the unit vector along M is: û_M = M/M. The unit vectors along x, y, and z a…

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If two vectors describing the same physical quantity are represented in magnitude and direction by the two sides of a triangle taken in order, then their resultant is represented in magnitude and dire…

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For vectors P and Q: P + Q = Q + P. Using triangle law, both P + Q and Q + P result in the same diagonal of a parallelogram formed by P and Q as adjacent sides. Triangle OAB shows P + Q = R, while tri…

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If two vectors of the same type, originating from the same point, are represented by two adjacent sides of a parallelogram, their resultant is given by the diagonal starting from the same point. Magni…

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Given: Rx = 3, Ry = 4. Magnitude: R = √(Rx² + Ry²) = √(3² + 4²) = √(9 + 16) = √25 = 5. Direction: θ = tan⁻¹(Ry/Rx) = tan⁻¹(4/3) = 53.13° with respect to x-axis.

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For vector R making angle θ with x-axis: Rx = R cos θ (x-component), Ry = R sin θ (y-component). In 3D: R = Rx î + Ry ĵ + Rz k̂, where Rx, Ry, Rz are components along x, y, z axes respectively.

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The scalar product of two vectors P and Q is defined as: P·Q = PQ cos θ, where θ is the angle between the vectors. It equals the product of magnitude of one vector with the component of the other vect…