0580 · 7.2Vectors in two dimensions

Vectors and translations

0:00Imagine moving a marker four squares right and three squares down on a grid. Someone else can repeat that movement from a different starting point. The destinations differ, but the movement is identical. A vector captures that movement, without needing to specify where it begins. By the end, you will understand how vectors describe translations and how to calculate with them. You need to know how coordinates work and how to calculate with negative numbers.

0:24The horizontal coordinate is x, and the vertical coordinate is y. On an ordinary coordinate grid, positive horizontal movement goes right, while positive vertical movement goes up. Moving left or down gives a negative component. A vector is a quantity with magnitude and direction. For a movement, magnitude means the length of the displacement, while direction tells us which way it points. An arrow represents both ideas: its length shows the magnitude, and its arrowhead shows the direction.

0:54Calculating magnitude belongs in the next video; here, we focus on horizontal and vertical components. A component tells us how much of the movement happens along one axis. Our arrow has horizontal component four and vertical component negative three. We write this as the column vector with x component four and y component negative three. The horizontal component sits above the vertical component, inside brackets. We can call this movement vector p, or describe it as vector PQ.

1:23The order of those point names matters because vector PQ goes from P to Q. In the exam, vectors are printed in bold or with an arrow above the letters. A column vector gives the same information through its two components. A coordinate describes a position, but a translation vector describes a change in position. The column vector with x component four and y component negative three does not mean the destination is fixed.

1:48It means the x coordinate rises by four while y falls by three. Wherever we start, those changes stay the same. A translation moves every point of a shape by the same vector. Each vertex shifts horizontally and vertically by identical amounts. That is why the shape keeps its size and orientation. To describe a translation, give its column vector rather than merely naming the image coordinates.

Finding a translation vector

2:13For our first worked example, describe the translation taking P to Q. Point P has x equal to negative two and y equal to four. Point Q has x equal to two and y equal to one. We need the change from start to finish in each direction, not the coordinates themselves. The horizontal change is 2 − −2, which equals four. For the vertical change, 1 − 4 gives negative three.

2:38The answer is a translation by the column vector with x component four and y component negative three. This works because the starting coordinate plus the change must equal the finishing one. Reversing the journey reverses both components. Vector QP is therefore the column vector with x component negative four and y component three. It takes us four left and three up, undoing the original movement. Changing a single sign would reverse only part of the movement, so it would not take us back.

Adding and subtracting vectors

3:05Adding vectors means combining movements. Follow vector p, then follow vector q from the endpoint of the first arrow. The resulting vector goes directly from the original starting point to the final endpoint. This is called head to tail addition, because the second arrow starts at the head of the first. Suppose vector p moves three right and two up, while vector q moves one left and four up. Together, the horizontal parts give two right, while vertically we rise six.

3:33We therefore add matching components. So we combine the two top entries, then the bottom pair. In this second example, first calculate vector p plus vector q. Adding three and negative one gives horizontal component two. The vertical calculation is 2 + 4, giving six. Our sum is the column vector with x component two and y component six. Subtracting vector q means adding its opposite. Since vector q moves one left and four up, its opposite moves one right and four down.

4:05Following vector p with that opposite movement explains vector p minus vector q. We can calculate it by subtracting the corresponding components. For the horizontal component, 3 − −1 = 4. The vertical component comes from 2 − 4, which gives negative two. Vector p minus vector q therefore has x component four and y component negative two. Notice that subtraction changes the entire vector q, not just its top entry.

4:33Addition gives the same final displacement whichever movement you make first. Subtraction is different because reversing the order reverses the answer. Also, horizontal components combine only with each other, and likewise the vertical ones. Adding all four entries into one number would lose the information about direction.

Scalar multiplication and mixed operations

4:52Multiplying a vector by a scalar scales its movement. A scalar is an ordinary number, rather than a vector. Multiplying by three triples both components, so the movement is repeated end to end. Multiplication by one half gives half the movement, with both components halved. A positive scalar keeps the direction unchanged. A negative scalar reverses the direction as well as scaling the movement. For example, multiplying by negative two doubles the length and points the arrow the opposite way.

5:21Multiplication by zero produces the zero vector, meaning no movement in either direction. For our third example, vector r has x component negative four and y component six. We first calculate negative two times vector r. Multiplying negative two by negative four gives horizontal component eight. For the vertical component, −2 × 6 = −12. The answer has x component eight and y component negative twelve, so the movement is now right and down.

5:51To find one half of vector r, halve each original component. The horizontal calculation gives negative two, while halving six gives vertical component three. This second answer points left and up, like the original, but represents half the movement. These operations can appear together in an exam question. Work out each scalar multiple first, then add or subtract matching components. The methods must work without a calculator on Paper two, and they also apply on calculator Paper four.

6:20These integer and fraction calculations give exact answers, so no rounding is needed. Vector u has x component two and y component negative three, while vector v has x component negative four and y component one. Calculate three vector u − 2 vector v. Multiplying both components of vector u by three gives x component six and y component negative nine. Twice vector v has x component negative eight and y component two.

6:47Keep these intermediate vectors visible so that the final subtraction is clear. The horizontal calculation is 6 − −8, giving fourteen. For the vertical component, −9 − 2 = −11. The final answer is the column vector with x component fourteen and y component negative eleven. Writing the intermediate multiples helps prevent a common mistake: applying the scalar to only one component. The key idea is that a vector records a displacement, not a destination.

7:17Its two components describe horizontal and vertical changes, and their signs tell us the directions. Addition combines movements, while subtraction adds the opposite movement. Scalar multiplication scales both components, with a negative scalar also reversing direction.