 2a вектор

# 2a вектор

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. And it doesn’t matter which order we add them, we get the same result: Содержание статьи:

### 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: ab

## 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: The vector a is broken up into
the two vectors ax and ay

(We see later how to do this.) 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 = a + b

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 = v + −k

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:

|a| = √( x2 + y2 )

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

|b| = √( 62 + 82) = √( 36+64) = √100 = 10

A vector with magnitude 1 is called a Unit Vector.

## Vector vs Scalar

A scalar has magnitude (size) only.

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

## 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 = 3m = (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) How do we multiply two vectors together? There is more than one way! (Read those pages for more details.)

## 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 = a + b

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) ?

|w| = √( 12 + (−2)2 + 32 ) = √( 1+4+9) = √14

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:

From Polar Coordinates (r,θ) to Cartesian Coordinates (x,y) From Cartesian Coordinates (x,y) to Polar Coordinates (r,θ) x = r × cos( θ ) y = r × sin( θ ) r = √ ( x2 + y2 ) θ = tan-1 ( y / x ) ## 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? First convert from polar to Cartesian (to 2 decimals):

Sam’s Vector:

• x = r × cos( θ ) = 200 × cos(60°) = 200 × 0.5 = 100
• y = r × sin( θ ) = 200 × sin(60°) = 200 × 0.8660 = 173.21

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: (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:

• r = √ ( x2 + y2 ) = √ ( 184.852 + 88.362 ) = 204.88
• θ = tan-1 ( y / x ) = tan-1 ( 88.36 / 184.85 ) = 25.5°

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!

Источник: www.mathsisfun.com

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