Use standard notations for vectors (column vectors, i-j-k notation, position vectors)
Carry out addition and subtraction of vectors and multiplication by a scalar, and interpret these operations geometrically
Calculate the magnitude of a vector, and use unit vectors, displacement vectors, and position vectors
Understand the equation of a straight line in the form r = a + tb, and find the equation of a line given sufficient information
Determine whether two lines are parallel, intersect, or are skew, and find the point of intersection when it exists
Use the scalar product to solve problems involving lines and points (e.g., finding angles between lines, finding the foot of the perpendicular from a point to a line)
A vector is a quantity that has both magnitude (size) and direction. This is different from a scalar, which only has magnitude (like temperature or mass).
Ways to Write Vectors
There are several standard ways to write vectors, and you need to be comfortable with all of them:
In 2 Dimensions:
Column vector notation: (xy) — The top number is the x-component, the bottom is the y-component
i-j notation: xi+yj — Here, i is a unit vector (length 1) pointing along the x-axis, and j is a unit vector pointing along the y-axis
In 3 Dimensions:
Column vector notation: xyz — Three components: x, y, and z
i-j-k notation: xi+yj+zk — Now we add k, a unit vector pointing along the z-axis
Position Vectors and Line Segments:
AB means the vector from point A to point B
a or a (bold) represents a general vector
OA or a represents the position vector of point A (the vector from the origin O to point A)
Example:
The vector (3−2) means "move 3 units in the x-direction and 2 units down in the y-direction."