î = (1)(1)(cos 90°) = 0ġ5 AB = AxBx + AyBy + AzBz If A & B are two vectors, whereĬase 1, (No angle θ) If A & B are two vectors, where A = Axi + Ayj + Azk & B = Bxi + Byj + Bzk Then, their Scalar Product is defined as: AB = AxBx + AyBy + AzBzġ6 Derivation How do we show that Start with Then But So. In addition, since a vector has no projection perpendicular to itself, the dot product of any unit vector with any other is zero. The scalar product of two vectors is written as It is also called the dot product q is the angle between A and B February 18, 2011ġ4 Scalar Product Applying this corollary to the unit vectors means that the dot product of any unit vector with itself is one. Why a Dot Product? – Because we use the notation A.Bġ1 Scalar Product of Two Vectors or (Dot product)ġ2 Scalar Product of Two Vectors is “Product of their magnitudes”. The second is called vector product or cross product because the result is a vector perpendicular to the plane of the two vectors.ġ0 Why Scalar Product? – Because the result is a scalar (just a number) The first one is called scalar product or dot product because the result of the product is a scalar quantity. Tan θ = Cy/Cx = 8/6 = 1.333, so we find θ = 53.1 degreeĩ There are two kinds of vector product : Solution, a- We know C = A + B Then, C = (Ax +Bx) (Ay +By) Then, C =( ) (1 + 7 ) = Thus, Cx = 6 & Cy = 8 b- From the Pythagorean theorem, C2 = Cx2 + Cy C2 = C = 10. X z y Consider 3D axes (x, y, z) i Define unit vectors, i, j, k j k Examples of Use: 40 m, E = 40 i m, W = -40 i 30 m, N = 30 j m, S = -30 j 20 m, out = 20 k 20 m, in = -20 kĦ Important Rule If A = Ax + Ay and B = Bx + By Then, C = A + Bħ Example, If A = & B = a- Find component C ( C = A + B) b- Find the magnitude of C and its angle with the x-axis. Unit Vector Notation, consider 2D axes(x, y) J = vector of magnitude in the “y” direction i = vector of magnitude in the “x” direction The hypotenuse is VECTOR SUM 3j Vertical Component =3j 4i Horizontal Component = 4i 
|i^|=1 and |j^|=1, this in two dimensions, and motion in three dimensions with ˆk (“k hat”) as the unit vector in the z directionĤ Unit Vector Notation, consider 2D axes(x, y)
Traditionally i^ (read “i hat”) is the unit vector in the x direction and j^ (read “j hat”) is the unit vector in the y direction. Zyad Ahmed TawfikĢ Lecture No.2 Unit Vector Notation Vectorsģ Unit Vector Notation, A unit vector is a vector that has a magnitude of one unit and can have any direction.
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