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MCV4U1 Vector Products

Today I will be talking about Dot and Cross products that we use in calculus. Lets first begin with on why we need these products in the first place. A dot product is an algebraic operation that takes two vectors and tells you the amount of one vector goes into another.

The basic formula is very simple since the format of the vectors are I, J, K, x,y,z are re presentable of them. For example I is Ax, J is Ay, and K is Az, the same goes for B. To find the dot product you multiply Ax and Bx and add the product of Ay and By and Az and Bz. After you find the dot product you can then use it to find the angle between the vectors. For example if A=(8,1) and B=(9,3) you would plug in these numbers into the formula and get, 8(9) + 1(3) = 72 + 3 = 75 so the dot product of A and B in this case is 75.

The formula is as shown cos θ = (a . b )/ |a||b|. What the denominator is the magnitude of a and b multiplying each other, the formula for that is every number that A and B are equal to will be squared and add on each other then square rooted. For example we will use A=(3,4) and B=(6,8), so it would be |a| = √(3^2)+(4)^2 = √25 = 5 and |b|= √(6^2)+(8^2) = √100 = 10. However in some cases there will be numbers that cannot be squared so for example we may have a situation like this √65 and √82 so what we do is multiply them both and that's our answer so it would look like this √5330. after that we can find the angle between the two vector with the dot product and the magnitude of the two vectors multiplied then divided by the dot product.

The Cross product is much different than the dot product and has its own uses it's formula is more complex than a dot product one. The steps for this product is, you have 3 values for a and b and you form them into a table like shown above, then for i, j, and k they will take every number but their own so for example i is equal to 1 and 2 but it will take 3, 4 7, and 5. then cross multiply them as shown above then sum the cross multiple of those numbers and the result is i. The same goes for j and k. Well why should we use cross product over dot product? There are different situations on when we use them. An example is the result of a dot product is a scalar quantity while the result of a cross product is a vector quantity.

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