what is the dot product between two different vectors

what is the dot product between two different vectors -...

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what is the dot product between two different vectors? The important thing to remember is that whatever we define the general rule to be, it must reduce to whenever we plug in two identical vectors. In fact, Equation has already been written suggestively to indicate that the general rule for the dot product between two vectors u = (u 1 , u 2 , u 3 ) and v = (v 1 , v 2 , v 3 ) might be: u·v = u 1 v 1 + u 2 v 2 + u 3 v 3 This equation is exactly the right formula for the dot product of two 3-dimensional vectors. (Note that the quantity obtained on the right is a scalar, even though we can no longer say it represents the length of either vector.) For 2-dimensional vectors, u = (u 1 , u 2 ) and v = (v 1 , v 2 ) , we have: u·v = u 1 v 1 + u 2 v 2 Again, by plugging in u = v , we recover the square of the length of the vector in two dimensions. Geometric Method
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So what does the scalar obtained in doing the dot product u.v represent? We can get an idea of what's going on by looking at the dot product of a vector with unit vectors. In Unit Vectors we defined the unit vectors i , j , and k for the 3-dimensional case. In two dimensions we have only i = (1, 0) and j = (0, 1) . (For now we will work in two dimensions, since it is easier to represent such vectors graphically.) The dot products of a vector v = (v 1 , v 2 ) with unit vectors i and j are given by: i = v 1 1 + v 2 0 = v 1 j = v 1 0 + v 2 1 = v 2 In other words, the dot product of v with i picks off the component of v in the x -direction, and similarly v 's dot product with j picks off the component of v which lies in the y -direction. This is the same as computing the magnitude of the projection of v onto the x - and y -axes, respectively. This may not seem too exciting, since in some sense we already knew this as soon as we wrote our vector down in terms of components. But what would happen if instead of components we were given only the direction and magnitude of a vector v , as in the following picture?
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Figure %: A vector v with length l and direction given by the angle θ . In this case, by noticing the two right triangles formed and recalling rules from trigonometry, we find that v· i and v· j can be computed in a different way. Namely: i = | v| cosθ j = | v| sinθ = l cos(90 - θ) What happens if we take the dot product of v with a generic vector which lies purely in the x -direction (i.e. not necessarily a unit vector)? We can write such a vector as w = (w 1 , 0) = w 1 (1, 0) = w 1 i , and it is clear that the magnitude of w is | w| = w 1 . Hence, w = | w| i . Using the above rule for the dot product between v and i , we find that: v·w = | v|| w| cosθ
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In fact, this equation holds in general: if we take v and w to be arbitrary vectors in either two or three dimensions, and let θ be the angle between them, we find that this version of the dot product formula agrees exactly with the component formula we found previously. Geometrically, the dot product v·w is given by | v|| w| cosθ .
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