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Three or More dimensions This all works fine in 3 (or more) dimensions, too. And can actually be very useful! Example: Sam has measured the end-points of two poles, and wants to know the angle between them : we have 3 dimensions, so don't forget the z-components: a b study ax bx ay by az bz a b a b a b 122 Now for the. It is the magnitude, or length, of the vector. We can use pythagoras : a ( ) a ( ) a 180 likewise for b : b ( ) b (81 4 49) b 134 And we know from the calculation above that a b 122, so: a b a b cos(θ) cos(θ) cos(θ). I tried a calculation like that once, but worked all in angles and distances. It was very hard, involved lots of trigonometry, and my brain hurt. The method above is much easier. Cross Product The dot Product gives a scalar (ordinary number) answer, and is sometimes called the scalar product.
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A vector has magnitude (how long it is) and direction : Here are two vectors: They can be multiplied using the dot Product " (also see, cross Product ). 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: a b a b cos(θ where: a is the magnitude (length) of vector. Example: Calculate the dot product of vectors a and b : a b a b cos(θ) a b cos(59.5).5075. 66 (rounded) or we can calculate it this way: a b ax bx ay by a b a b a b 66 Both methods came up with the same result qualitative (after rounding) Also note that we used minus 6 for ax (it is heading. Ok, to multiply two vectors it makes sense to multiply their lengths together but only when they point in the same direction. So we make one "point in the same direction" as the other by multiplying by cos(θ we take the component of a that lies alongside b like shining a light to see where the shadow lies then we multiply! It works exactly the same if we "projected" b alongside a then multiplied: Because it doesn't matter which order we do the multiplication: a b cos(θ) a cos(θ) b right Angles When two vectors are at right angles to each other the dot product. Example: calculate the dot Product for: a b a b cos(θ) a b a b cos(90) a b a b 0 a b 0 or we can calculate it this way: a b ax bx ay by a b a b a b 0 This.