The super easy and quick concept of Dot product of vectors or dot multiplication of vectors or scalar multiplication of vectors is explained in this video.
The video covers all about the dot product and scalar produc of vectors. I will also tell you about:
1) Dot Product and Scalar Product of Vectors, Physics.
2) Physics - Mechanics: Vectors (12 of 21) Product Of Vectors: Dot Product
3) The Dot Product.
4) Class 12 Maths : Vectors - Dot product ( Scalar product ) Part 1.
5) Scalar Product or Dot Product (Hindi).
6) Cross Product and Dot Product: Visual explanation.
7) Dot vs. cross product | Physics | Khan Academy
Dot product of vectors or scalar product of vectors is the multiplication of two or more vectors. Dot product or scalar product of two vectors always give us a number.
a scalar function of two vectors, equal to the product of their magnitudes and the cosine of the angle between them.
The Dot Product gives a number as an answer (a "scalar", not a vector).
The Dot Product is written using a central dot:
a · b
This means the Dot Product of a and b
We can calculate the Dot Product of two vectors this way:
dot product magnitudes and angle
a · b = |a| × |b| × cos(θ)
|a| is the magnitude (length) of vector a
|b| is the magnitude (length) of vector b
θ is the angle between a and b
he dot product between two vectors is based on the projection of one vector onto another. Let's imagine we have two vectors aa and bb, and we want to calculate how much of aa is pointing in the same direction as the vector bb. We want a quantity that would be positive if the two vectors are pointing in similar directions, zero if they are perpendicular, and negative if the two vectors are pointing in nearly opposite directions. We will define the dot product between the vectors to capture these quantities.
But first, notice that the question “how much of aa is pointing in the same direction as the vector bb” does not have anything to do with the magnitude (or length) of bb; it is based only on its direction. (Recall that a vector has a magnitude and a direction.) The answer to this question should not depend on the magnitude of bb, only its direction. To sidestep any confusion caused by the magnitude of bb, let's scale the vector so that it has length one. In other words, let's replace bb with the unit vector that points in the same direction as bb. We'll call this vector uu, which is defined by
The dot product of aa with unit vector uu, denoted a⋅ua⋅u, is defined to be the projection of aa in the direction of uu, or the amount that aa is pointing in the same direction as unit vector uu. Let's assume for a moment that aa and uu are pointing in similar directions. Then, you can imagine a⋅ua⋅u as the length of the shadow of aa onto uu if their tails were together and the sun was shining from a direction perpendicular to uu. By forming a right triangle with aa and this shadow, you can use geometry to calculate that
where θθ is the angle between aa and uu.
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