Vectors Multiplying Vectors By A Scalar Quantity Vectors

Vectors Multiplying Vectors By A Scalar Quantity Vectors are quantities that have both magnitude (size) and direction. The vector shown can be written as a = 4 i +2 j If we want a vector twice as long as a in the same direction, we can multiply a by 2. 2 is a scalar quantity. The length of the vector is altered but the direction stays the same. 2 a The new vector is 2 a = 2 ( 4 i + 2 j ) a 2 4 2 a = 8 i + 4 j Multiplying by a scalar has the effect of multiplying each of the components by the scalar number

Vectors Multiplying Vectors By A Scalar Quantity The vector shown can be written as a = 4 i +2 j The negative vector –a has the same length as a but the opposite direction. When a is given in component form, the components of –a are the same as those for a but with the signs reversed. The new vector is –a = – ( 4 i + 2 j ) –a a 2 4 –a = – 4 i – 2 j

Vectors Adding Vectors When vectors are in component form, they can be added component by component. Add the vectors 3 i + 5 j & 6 i – j 3 i + 5 j 9 i + 4 j 3 i + 5 j + 6 i – j = 9 i + 4 j The sum of two or more vectors is called the resultant, and is usually represented by a double arrowhead. 9 i + 4 j is the resultant vector.

Vectors Resultant Vectors In mechanics, vectors are used to represent forces acting on a body. We can add together all the forces to find the resultant force acting on the object. Diagram not to scale. F 2 12 N 10 N 60˚ 50 N F 1 F 3 Find the resultant force acting on this object. F 1 + F 2 + F 3 =