Vector introduction PDF

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Introduction to vector calculations....

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Vector Introduction Vectors are physical quantities with magnitude and direction. Some examples include; velocity, acceleration, displacement, force, torque, momentum, angular velocity and vorticity. Vectors can be expressed in many ways: •

Geometrically: An arrow represents its direction, and its relative length is its magnitude (e.g. velocity in that direction, with speed x ms-1). x

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Numerically: In relation to a set of axes (usually x, y and z), where a, b and c are the three components of v. Then i, j and k ae the base vectors (length of one) along each axis. v = ai + bj + ck

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Another notation: Sometimes a bracket or matrix notation can be used to display a vector too. 𝑎 v = ( a, b, c) or v = ! 𝑏 % 𝑐

A vector is always underlined, you will not gain any marks if this is not done. To work out the magnitude of a vector (the ‘length of the arrow’) then you need to use Pythagoras’ theorem to calculate this. |v|= &(𝑎( + 𝑏( + 𝑐 ( ) e.g. F=2i-j-2k à |F|=3N Unit vectors have a magnitude of 1 (|v|=1) and it is common to denote these unit vectors by using circumflex (e.g. â) above the letter. A unit vector + can be constructed by doing ñ= | | (𝑎𝑖 + 𝑏𝑗 + 𝑐𝑘). You can scale vectors as normal by multiplying all the coefficients by a specific value, e.g. v=2i-3j+4k is the same as 2v=4i-6j+8k. If a=a1i+a2j+a3k (a1,a2,a3) and b=b1i+b2j+b3k (b1,b2,b3) then axb=a1b1+ a2b2+ a3b3 or in matrix form: 𝑏1 (𝑎1 𝑎2 𝑎3) 4 𝑏25=abT 𝑏3 To work out the angle between the vectors, use the formula below and note that if 𝜃=90º then a.b=0… 𝑎;. 𝑏 cos 𝜃 = |𝑎|⌈𝑏⌉ Given the vectors a and b, the product ab is another vector which is perpendicular to a and b and has a magnitude of |a||b|sin 𝜃.

ab points in the direction such that rotation from a to b is clockwise. This means that ba=-ab. Note that ij=k, jk=i and ki=j Given the same values for vectors a and b above (where a=a1i+a2j+a3k (a1,a2,a3) and b=b1i+b2j+b3k (b1,b2,b3)) then there is a pattern for working out what ab will be. ab= (a2b3-a3b2)i - (a1b3-a3b1)j + (a1b2-a2b1)k . To remember this pattern you can write it out in full and work out the determinants for each matrix-style vector: 𝑖 𝑗 𝑘 𝑎2 𝑎3 𝑎1 𝑎3 𝑎1 𝑎2 𝑎𝑏 = 4 𝑎1 𝑎2 𝑎35 = 𝑖 A B − 𝑗A B + 𝑘A B 𝑏2 𝑏3 𝑏1 𝑏3 𝑏1 𝑏2 𝑏1 𝑏2 𝑏3 = ; (a2b3 − a3b2)i; − ; (a1b3 − a3b1)j; + ; (a1b2 − a2b1)k For example: a=i-3j+2k and b=-i-j+k . Find c, the product of a.b: 𝑖 𝑐 = 𝑎𝑏 = 4 1 −1

𝑗 𝑘 −3 2 1 2 1 −3 −3 25 = 𝑖 A−1 1B − 𝑗 A−1 1B + 𝑘 A−1 −1B −1 1 c = (−3 + 2)i − (1 + 2)j + (−1 − 3)k c = ; −i − 3j + 4k...