Vectors in two dimensions.
Describe a displacement with two components, then combine and scale the directed steps.
Use column and directed-vector notation · add and subtract vectors · multiply by a scalar
A vector has size and direction
A position tells you where something is; a displacement tells you how it moved. A vector describes displacement with both magnitude and direction. The same movement can start at different locations and still be the same vector.
In a two-component column, the top number is the horizontal change and the bottom number is the vertical change. This is not a fraction. To find the vector from A to B, subtract A’s coordinates from B’s coordinates, keeping the destination-minus-start order.
A vector describes a displacement, not a fixed position. Equal vectors have the same components, magnitude and direction wherever they are drawn.
This means 3 units right and 2 units down. The same displacement can be written as a bold vector a or a directed segment AB from its starting point A to its endpoint B.
A = (−2, 4) and B = (3, 1). Find vector AB.
Vector BA is the opposite: −53. Reversing an arrow negates the vector.
Pause and explainAre vectors describing 3 right and 2 up different if they start at different points?
No. They represent the same displacement because their components, magnitude and direction match.
Add head to tail; subtract by reversing
If you walk east and then north, the combined displacement runs directly from your original starting point to your final position. Vector addition describes that combined movement.
Draw the second arrow starting at the head of the first, without changing its length or direction. The resultant joins the first tail to the final head. Subtraction adds the opposite movement, so reverse the vector being subtracted before constructing the route.
Let a = 31 and b = −13. Find their sum and difference.
a − b = 3 − (−1)1 − 3 = 4−2
Subtract component by component. Geometrically, a − b = a + (−b).
Pause and explainWhat displacement results from adding a vector to its negative?
The zero vector. The second movement exactly reverses the first, returning to the starting point.
A scalar multiplies every component
A scalar is an ordinary number used to scale a vector. Multiplying a movement by 3 makes both its horizontal and vertical changes three times as large.
A positive scalar preserves direction, while a negative one reverses it. The absolute scalar gives the length multiplier. Multiplying by zero removes the whole movement, leaving a vector with zero magnitude and no defined direction.
A positive scalar keeps direction and multiplies length by that scalar.
If a = 2−3 and b = −14, find 3a − 2b.
Pause and explainWhy must both components change when a vector is multiplied by −2?
The whole displacement is reversed and doubled. Scaling only one component would generally produce a different direction.
Your turn. Work one component at a time.
Write column components neatly and keep horizontal calculations separate from vertical ones. For a directed vector, identify the starting and ending points before subtracting.
Sketch a head-to-tail route when addition or subtraction is hard to picture. After scalar multiplication, check both the numerical components and whether the direction should stay the same or reverse.
Find 4−1 + −65.
Reveal worked answer
Find 2−3 − −41.
Reveal worked answer
Find −3−25.
Reveal worked answer
A = (3, −1), B = (−2, 7). Find AB and BA.