scalars and vectors
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- Last Updated: Monday, 07 September 2026 23:27
- Published: Sunday, 06 September 2026 21:29
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scalars and vectors:
This topic is covered in Physics as well as in maths (in calculus). I'm covering in Physics as most of it's application is in Physics.
Scalars: Things which don't have any sense of direction, are called scalars. As an example of scalar, mass of a thing, is a scalar quantity. It doesn't have any sense of direction, whether it's going north, south, etc, the mass is the same.
Vectors: Things which have direction, are called vectors. As an example of vector, force is a vector quantity. It depends on which direction you apply force, so it has a sense of direction. IF the force is applied in same direction as the movement of the body, then the body will accelerate, while if it's applied in opposite dir, then the body slows down.So, direction of force matters.
Vector: A vector is rep by an arrow, whose length is the magnitude of the vector, and whose dir is the angle that that vector makes wrt x and y axis (for 2D vector), or wrt x, y and z axis (for 3D vector). Vectors can be moved anywhere in space and will remain same, as long as mag and dir remain same (i.e starting point and ending pt of vector can be anywhere).
Vector is rep by i, j, k for x,y and z dir or by coords (x,y,z) or in matrix form as x, y, z written vertically in 1st column.
We'll study some operations (algebra) on vector quantities.
- Addition/subtraction: Addition/subtraction of vectors is done by placing end point of a vector to the start point of next vector and so on, until all vectors are connected. Then you connect the start point of 1st vector to end point of last vector, and that's the sum of all the vectors. For subtraction, you still treat it as addition, except that the direction of vector is reversed for the subtracting vectors.
- For vector <x1,y1> and <x2,y2>, the sum is <x1+x2,y1+y2>. The magnitude of vector is sq rt of [(x1+x2)^2 + (y1+y2)^2]. Mag may also be calc by using cosine thm,
- Multiplication: There's no multiplication of vectors. We call it the product of vectors. Product of vectors may give a scalar or a vector depending on how we define it. There are 2 kinds of product. Link => https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Vector_Calculus/1%3A_Vector_Basics/1.5%3A_The_Dot_and_Cross_Product
- Dot product: It's a scalar, and is determined by multiplying coefficients as shown in link above.
- Dot product is simply multiplying the length of one vector, with the length of the shadow (or projection) of the other vector on the 1st vector. So, u.v = ||u||.||v cos θ|| = ||u||.||v|| cos θ. Ot can also be calculated by using formula u.v=u1.v1 + u2.v2 + u3.v3.
- Cross product: It's a vector, and is determined by multiplying coefficients as shown in link above. It also has dir (as the product is a vector).
- Cross product is found by getting determinant of the matrix with alternating +ve and -ve sign. It also has dir which is perpendicular to the plane of the 2 vectors u, v.
- Dot product: It's a scalar, and is determined by multiplying coefficients as shown in link above.
Matrix: Matrices are useful when evaluating dot and cross product of matrices.