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MathQuiz: on-line quizzes written using LaTeX (and TeX4ht)

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We will find \overrightarrow{EB} by doing

\overrightarrow{EB}=\overrightarrow{EA}+\overrightarrow{AB}

The first vector is straightforward, because we know \overrightarrow{AE}, and that is just the same vector in the opposite direction. So, we get:

\overrightarrow{EA}=-\overrightarrow{AE}=-(3\mathbf{a}-2\mathbf{b})=-3\mathbf{a}+2\mathbf{b}

Now we need \overrightarrow{AB}. Since B is the midpoint of AC (given in the question), we must have that \overrightarrow{AB}=\frac{1}{2} \overrightarrow{AC}. Therefore, looking at the diagram, we get that:

\overrightarrow{AB}=\dfrac{1}{2}\overrightarrow{AC}=\dfrac{1}{2}\left(\overrightarrow{AD}+\overrightarrow{DC}\right)

We're given the second part of this, \overrightarrow{DC}=2\mathbf{a}+4\mathbf{b}, and since E is the midpoint of AD, we can also work out the first part:

\overrightarrow{AD}=2\overrightarrow{AE}=2(3\mathbf{a}-2\mathbf{b})=6\mathbf{a}-4\mathbf{b}

Now, we have everything we need and can go back through our work, filling in the gaps we find:

\overrightarrow{AB}=\dfrac{1}{2}\left(6\mathbf{a}-4\mathbf{b}+2\mathbf{a} +4\mathbf{b}\right)=\dfrac{1}{2}\left(8\mathbf{a}\right)=4\mathbf{a}

Finally we have that:

\overrightarrow{EB}=\overrightarrow{EA}+\overrightarrow{AB}=-3\mathbf{a}+2\mathbf{b}+4\mathbf{a}=\mathbf{a}+2\mathbf{b}

If \overrightarrow{EB} and \overrightarrow{DC} are parallel, then one must be a multiple of the other. Well, if we multiply \overrightarrow{EB} by 2 then,

2\times\overrightarrow{EB}=2(\mathbf{a}+2\mathbf{b})=2\mathbf{a}+4\mathbf{b}=\overrightarrow{DC}

We have shown that 2\overrightarrow{EB}=\overrightarrow{DC}, therefore, the two lines must be parallel.

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Vectors Questions | Worksheets and Revision | MME

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We will find \overrightarrow{EB} by doing \overrightarrow{EB}=\overrightarrow{EA}+\overrightarrow{AB} The first vector is straightforward, because we know \overrightarrow{AE}, and that is just the same vector in the opposite direction. So, we get: \overrightarrow{EA}=-\overrightarrow{AE}=-(3\mathbf{a}-2\mathbf{b})=-3\mathbf{a}+2\mathbf{b} Now we need \overrightarrow{AB}. Since B is the midpoint of AC (given in the question), we must have that \overrightarrow{AB}=\frac{1}{2} \overrightarrow{AC}.

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A vector describes a movement from one point to another. A vector quantity has both direction and magnitude (size).

A scalar quantity has only magnitude.

A vector can be represented by a line segment labelled with an arrow.

A vector between two points A and B is described as: , or .

The vector can also be represented by the column vector . The top number tells you how many spaces or units to move in the positive -direction and the bottom number is how many to move in the positive -direction.

Vectors are equal if they have the same magnitude and direction regardless of where they are.

So

A negative vector has the same magnitude but the opposite direction.

Vector is the same as travelling backwards down the vector .

Example

Write, in terms of , and , the vectors , , and .

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BBC Bitesize - GCSE Maths - Vectors - AQA - Revision 1

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A vector describes a movement from one point to another. A vector quantity has both direction and magnitude (size). A scalar quantity has only magnitude. A vector can be represented by a line segment labelled with an arrow. A vector between two points A and B is described as: , or .

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This is a vector:

A vector has magnitude (size) and direction:

The length of the line shows its magnitude and the arrowhead points in the direction.

We can add two vectors by joining them head-to-tail:

And it doesn't matter which order we add them, we get the same result:

Example: A plane is flying along, pointing North, but there is a wind coming from the North-West.

The two vectors (the velocity caused by the propeller, and the velocity of the wind) result in a slightly slower ground speed heading a little East of North.

If you watched the plane from the ground it would seem to be slipping sideways a little.

Have you ever seen that happen? Maybe you have seen birds struggling against a strong wind that seem to fly sideways. Vectors help explain that.

Velocity, acceleration, force and many other things are vectors.

Subtracting

We can also subtract one vector from another:

first we reverse the direction of the vector we want to subtract,

then add them as usual:

a − b

Notation

A vector is often written in bold, like a or b.

A vector can also be written as the letters
of its head and tail with an arrow above it, like this:

Calculations

Now ... how do we do the calculations?

The most common way is to first break up vectors into x and y parts, like this:

(We see later how to do this.)

Adding Vectors

We can then add vectors by adding the x parts and adding the y parts:

The vector (8, 13) and the vector (26, 7) add up to the vector (34, 20)

Example: add the vectors a = (8, 13) and b = (26, 7)

c = (8, 13) + (26, 7) = (8+26, 13+7) = (34, 20)

When we break up a vector like that, each part is called a component:

Subtracting Vectors

To subtract, first reverse the vector we want to subtract, then add.

Example: subtract k = (4, 5) from v = (12, 2)

a = (12, 2) + −(4, 5) = (12, 2) + (−4, −5) = (12−4, 2−5) = (8, −3)

Magnitude of a Vector

The magnitude of a vector is shown by two vertical bars on either side of the vector:

| a|

OR it can be written with double vertical bars (so as not to confuse it with absolute value):

|| a||

We use Pythagoras' theorem to calculate it:

Example: what is the magnitude of the vector b = (6, 8) ?

A vector with magnitude 1 is called a Unit Vector.

Vector vs Scalar

Scalar: just a number (like 7 or −0.32) ... definitely not a vector.

A vector has magnitude and direction, and is often written in bold, so we know it is not a scalar:

so c is a vector, it has magnitude and direction

but c is just a value, like 3 or 12.4

Example: k b is actually the scalar k times the vector b.

Multiplying a Vector by a Scalar

When we multiply a vector by a scalar it is called "scaling" a vector, because we change how big or small the vector is.

Example: multiply the vector m = (7, 3) by the scalar 3

a = 3 m = (3×7, 3×3) = (21, 9)

It still points in the same direction, but is 3 times longer

(And now you know why numbers are called "scalars", because they "scale" the vector up or down.)

Multiplying a Vector by a Vector (Dot Product and Cross Product)

More Than 2 Dimensions

Vectors also work perfectly well in 3 or more dimensions:

The vector (1, 4, 5)

Example: add the vectors a = (3, 7, 4) and b = (2, 9, 11)

c = (3, 7, 4) + (2, 9, 11) = (3+2, 7+9, 4+11) = (5, 16, 15)

Example: what is the magnitude of the vector w = (1, −2, 3) ?

Here is an example with 4 dimensions (but it is hard to draw!):

Example: subtract (1, 2, 3, 4) from (3, 3, 3, 3)

(3, 3, 3, 3) + −(1, 2, 3, 4)
= (3, 3, 3, 3) + (−1,−2,−3,−4)
= (3−1, 3−2, 3−3, 3−4)
= (2, 1, 0, −1)

Magnitude and Direction

We may know a vector's magnitude and direction, but want its x and y lengths (or vice versa):

<=>

Vector a in Polar
Coordinates

Vector a in Cartesian
Coordinates

You can read how to convert them at Polar and Cartesian Coordinates, but here is a quick summary:

An Example

Sam and Alex are pulling a box.

Sam pulls with 200 Newtons of force at 60°

Alex pulls with 120 Newtons of force at 45° as shown

What is the combined force, and its direction?

Let us add the two vectors head to tail:

First convert from polar to Cartesian (to 2 decimals):

Sam's Vector:

Alex's Vector:

x = r × cos( θ ) = 120 × cos(−45°) = 120 × 0.7071 = 84.85

y = r × sin( θ ) = 120 × sin(−45°) = 120 × -0.7071 = −84.85

Now we have:

Add them:

(100, 173.21) + (84.85, −84.85) = (184.85, 88.36)

That answer is valid, but let's convert back to polar as the question was in polar:

And we have this (rounded) result:

And it looks like this for Sam and Alex:

They might get a better result if they were shoulder-to-shoulder!

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This is a vector: A vector has magnitude (size) and direction: The length of the line shows its magnitude and the arrowhead points in the direction.

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Vectors GCSE Maths revision section looking at Vectors, including diagrams and examples.

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Vectors | Chapter 1, Essence of linear algebra