Unit Vectors. Old school ► Consider vector B, which is expressed as B and θ where B is the magnitude and θ is the direction ► We know that the vector.

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Presentation transcript:

Unit Vectors

Old school ► Consider vector B, which is expressed as B and θ where B is the magnitude and θ is the direction ► We know that the vector has components in the x and y axis (every vector does) B x and B y ► We can find these components using trig functions and pythag theorem

► We always MUST consider the x and y components of a vector to do any type of physics or math problems

New School ► Vectors can also be expressed in terms of unit vectors ► A unit vector is a dimensionless vector having a magnitude of exactly 1  The unit vector is only used to specify a given direction  i, j, k are used to represent unit vectors pointing in the direction of the positive x, y, z directions

► If we go back to vector B, now we can express it as B at θ = B x i + B y j B at θ = B x i + B y j since Bx is in the i direction since Bx is in the i direction and By is in the j direction and By is in the j direction So any position vector R can be written as R = R x i + R y j (+ R z k) Ex. If R = o, it can be broken into x=12.29 m and y=8.6 m so… R = i j m R = i j m

Adding vectors in unit vector form becomes very easy A + B, where A = A x i + A y j, B = B x i + B y j So if A + B = C, then C = (A x + B x ) i + (A y + B y ) j And once we have C, then we can still find it’s polar coordinates, or magnitude and direction (pythag & trig)

► Sometimes we will do 3 dimensional problems. You can actually still use pythag theorem to find the resultant magnitude, but never anything with angles (at least for us) ► You can add (we saw this), subtract (add opposite), and multiply unit vectors, just like before ex. Find 5A where A = A x i + A y j 5A = 5A x i + 5A y j 5A = 5A x i + 5A y j

Find the sum of two vectors a) in unit vector notation and b) polar coordinates if A = 2i + 2j and B = 2i – 4j A = 2i + 2j and B = 2i – 4j

A particle undergoes three consecutive displacements: d 1 = 15i + 30j + 12k cm d 2 = 23i – 14j – 5k cm d 3 = -13i + 15j cm d 3 = -13i + 15j cm Find the resultant magnitude of this displacement.