Mathematics 1

Vectors

Definitions

• A scalar is a quantity which has magnitude only, e.g., temperature, mass, pressure.
• A vector is a quantity which has both: a magnitude and a direction e.g., displacement,
• velocity, magnetic field.
• The directed line segment
• -represents a vector
• The length PQ- its magnitude
• Notations of vectors: ; ; ;
• Notations of magnitudes:|a| or ||a||

•  

P
Q

The triangle rule.

We can add two vectors by joining them head-to-tail:
it doesn't matter which order we add them, we get the same result:
This rule for adding vectors represented by line segments is called the triangle rule.
a + b = b + a

Subtraction of vectors

• -A is defined to be the vector with the same magnitude as A but opposite direction.
• 0 is the zero vector (or null vector) with zero magnitude and any direction.

To subtract, first reverse the vector we want to subtract, then add using the triangle rule

Sample problem 1

Write as single vectors:

• f + g
• a + b
• e - b - a

Answer:

• e
• -c
• -d

Multiplication by scalars

• We can multiply vectors by scalars (i.e. numbers).
• When p > 0, pa is defined to be the vector

with the same direction as and p times the magnitude of a.

• When p < 0, pa is the vector with the opposite direction to a and -p = times the magnitude.
• Pictorially this looks like stretching or squeezing the vector.
• When we multiply a vector by a scalar it is called "scaling" a vector, because we change how big or small the vector is.
• A vector of magnitude one is