Как пишется угол между векторами

Два вектора

a→

и

b→

 всегда образуют угол.

Угол между векторами может принимать значения от

0°

до

180°

включительно.

4. угол величиной

0°

 (векторы сонаправлены);

5. угол величиной

180°

 (векторы противоположно направлены).

Если один из векторов или оба вектора нулевые, то угол между ними будет равен

0°

.

1. Если угол между векторами острый, то скалярное произведение будет положительным числом (так как косинус острого угла — положительное число). 

2. Если угол между векторами тупой, то скалярное произведение будет отрицательным (так как косинус тупого угла — отрицательное число). 

Справедливы и обратные утверждения:

1. Если скалярное произведение векторов — положительное число, то угол между данными векторами острый.

2. Если скалярное произведение векторов — отрицательное число, то угол между данными векторами тупой.

Обрати внимание!

Свойства скалярного произведения

Использование скалярного произведения

Вектор называют направляющим вектором прямой, если он находится на прямой или параллелен этой прямой.

Если

a→x1;y1;z1

,

b→x2;y2;z2

, то

a→⋅b→=x1⋅x2+y1⋅y2+z1⋅z2

.

Теперь, объединив эти формулы, получим формулу для определения косинуса угла между векторами в координатной форме. Так как из формулы скалярного произведения следует, что

cosα=a→⋅b→a→⋅b→

, то

.

Используя следующий рисунок, легко доказать, что косинус угла

β

между нормальным вектором

n→

 данной плоскости и неким вектором

b→

 равен синусу угла

α

между прямой и плоскостью, так как

α

и

β

 вместе образуют угол в

90°

.

При нахождении косинуса угла между

n→

и

b→

можно использовать это число как синус угла между прямой, на которой лежит вектор

b→

, и плоскостью.

Скалярное произведение двух данных векторов $overline{δ}$ и $overline{β}$ с координатами $(δ_1,β_1,γ_1)$ и $(δ_2,β_2,γ_2)$, равняется сумме произведений их соответствующих координат.

Математически выглядит следующим образом

$overline{δ}cdot overline{β}=δ_1 δ_2+β_1 β_2+γ_1 γ_2$

Найти косинус угла между векторами $overline{δ}$ и $overline{β}$, имеющими координаты $(1,-2,2)$ и $(3,0,4)$, соответственно.

Решение.

Найдем скалярное произведение между данными векторами через координаты:

$overline{δ}cdot overline{β}=1cdot 3+(-2)cdot 0+2cdot 4=11$

Найдем длины этих векторов:

$|overline{δ}|=sqrt{1^2+(-2)^2+2^2}=sqrt{9}=3$

$|overline{β}|=sqrt{3^2+0^2+4^2}=sqrt{25}=5$

В результате, получим

$cos⁡∠(overline{δ},overline{β})=frac{11}{3cdot 5}=frac{11}{15}$

Ответ: $frac{11}{15}$.

Найти синус угла между векторами $overline{δ}$ и $overline{β}$, имеющими координаты $(1,-2,2)$ и $(3,0,4)$, соответственно.

Решение.

Найдем вектор векторного произведения между данными векторами по формуле:

$overline{δ}хoverline{β}=begin{vmatrix}overline{i}&overline{j}&overline{k}\1&-2&2\3&0&4end{vmatrix}=-8overline{i}+2overline{j}+6overline{k}=(-8,1,6)$

Найдем длины этих векторов:

$|overline{δ}хoverline{β}|=sqrt{(-8)^2+2^2+6^2}=sqrt{104}=2sqrt{26}$

$|overline{δ}|=sqrt{1^2+(-2)^2+2^2}=sqrt{9}=3$

$|overline{β}|=sqrt{3^2+0^2+4^2}=sqrt{25}=5$

В результате, получим

$sin∠(overline{δ},overline{β})=frac{2sqrt{26}}{3cdot 5}=frac{2sqrt{26}}{15}$

Ответ: $frac{2sqrt{26}}{15}$.

1. 

   1

   Write down the cosine formula. To find the angle θ between two vectors, start with the formula for finding that angle’s cosine. You can learn about this formula below, or just write it down:^[1]
   

2. 

   2

   Identify the vectors. Write down all the information you have concerning the two vectors. We’ll assume you only have the vector’s definition in terms of its dimensional coordinates (also called components). If you already know a vector’s length (its magnitude), you’ll be able to skip some of the steps below.

   Advertisement

3. 

   3

   Calculate the length of each vector. Picture a right triangle drawn from the vector’s x-component, its y-component, and the vector itself. The vector forms the hypotenuse of the triangle, so to find its length we use the Pythagorean theorem. As it turns out, this formula is easily extended to vectors with any number of components.

4. 

   4

   Calculate the dot product of the two vectors. You have probably already learned this method of multiplying vectors, also called the scalar product.^[2]
   

   To calculate the dot product in terms of the vectors’ components, multiply the components in each direction together, then add all the results.

   For computer graphics programs, see Tips before you continue.

   Finding Dot Product Example
   In mathematical terms, • = u₁v₁ + u₂v₂, where u = (u₁, u₂). If your vector has more than two components, simply continue to add + u₃v₃ + u₄v₄…
   In our example, • = u₁v₁ + u₂v₂ = (2)(0) + (2)(3) = 0 + 6 = 6. This is the dot product of vector and .

5. 

   5

   Plug your results into the formula. Remember,

   cosθ = ( • ) / (|||| || ||).

   Now you know both the dot product and the lengths of each vector. Enter these into this formula to calculate the cosine of the angle.

   Finding Cosine with Dot Product and Vector Lengths
   In our example, cosθ = 6 / (2√2

   3) = 1 / √2 = √2 / 2.

6. 

   6

   Find the angle based on the cosine. You can use the arccos or cos^-1 function on your calculator to

   find the angle θ from a known cos θ value.

   For some results, you may be able to work out the angle based on the unit circle.

   Finding an Angle with Cosine
   In our example, cosθ = √2 / 2. Enter «arccos(√2 / 2)» in your calculator to get the angle. Alternatively, find the angle θ on the unit circle where cosθ = √2 / 2. This is true for θ = ^π/₄ or 45º.
   Putting it all together, the final formula is:
   angle θ = arccosine(( • ) / (|||| ||||))

Advertisement

1. 

   1

   Understand the purpose of this formula. This formula was not derived from existing rules. Instead, it was created as a definition of two vectors’ dot product and the angle between them. However, this decision was not arbitrary. With a look back to basic geometry, we can see why this formula results in intuitive and useful definitions.

   • The examples below use two-dimensional vectors because these are the most intuitive to use. Vectors with three or more components have properties defined with the very similar, general case formula.

2. 

   2

   Review the Law of Cosines. Take an ordinary triangle, with angle θ between sides a and b, and opposite side c. The Law of Cosines states that c² = a² + b² -2abcos(θ). This is derived fairly easily from basic geometry.

3. 

   3

   Connect two vectors to form a triangle. Sketch a pair of 2D vectors on paper, vectors and , with angle θ between them. Draw a third vector between them to make a triangle. In other words, draw vector such that + = . This vector = — .^[3]
   

4. 

   4

   Write the Law of Cosines for this triangle. Insert the length of our «vector triangle» sides into the Law of Cosines:

   • ||(a — b)||² = ||a||² + ||b||² — 2||a|| ||b||cos(θ)

5. 

   5

   Write this using dot products. Remember, a dot product is the magnification of one vector projected onto another. A vector’s dot product with itself doesn’t require any projection, since there is no difference in direction.^[4]
   This means that • = ||a||². Use this fact to rewrite the equation:

6. 

   6

   Rewrite it into the familiar formula. Expand the left side of the formula, then simplify to reach the formula used to find angles.

Advertisement

Advertisement

Video

• If you are working on a computer graphics program, you most likely only care about the direction of the vectors, not their length. Take these steps to simplify the equations and speed up your program:^{[5]
  [6]}
  

• For a quick plug and solve, use this formula for any pair of two-dimensional vectors: cosθ = (u₁ • v₁ + u₂ • v₂) / (√(u₁² • u₂²) • √(v₁² • v₂²)).

Did this summary help you?

1. 

   1

   Write down the cosine formula. To find the angle θ between two vectors, start with the formula for finding that angle’s cosine. You can learn about this formula below, or just write it down:^[1]
   

2. 

   2

   Identify the vectors. Write down all the information you have concerning the two vectors. We’ll assume you only have the vector’s definition in terms of its dimensional coordinates (also called components). If you already know a vector’s length (its magnitude), you’ll be able to skip some of the steps below.

   Advertisement

3. 

   3

   Calculate the length of each vector. Picture a right triangle drawn from the vector’s x-component, its y-component, and the vector itself. The vector forms the hypotenuse of the triangle, so to find its length we use the Pythagorean theorem. As it turns out, this formula is easily extended to vectors with any number of components.

4. 

   4

   Calculate the dot product of the two vectors. You have probably already learned this method of multiplying vectors, also called the scalar product.^[2]
   

   To calculate the dot product in terms of the vectors’ components, multiply the components in each direction together, then add all the results.

   For computer graphics programs, see Tips before you continue.

   Finding Dot Product Example
   In mathematical terms, • = u₁v₁ + u₂v₂, where u = (u₁, u₂). If your vector has more than two components, simply continue to add + u₃v₃ + u₄v₄…
   In our example, • = u₁v₁ + u₂v₂ = (2)(0) + (2)(3) = 0 + 6 = 6. This is the dot product of vector and .

5. 

   5

   Plug your results into the formula. Remember,

   cosθ = ( • ) / (|||| || ||).

   Now you know both the dot product and the lengths of each vector. Enter these into this formula to calculate the cosine of the angle.

   Finding Cosine with Dot Product and Vector Lengths
   In our example, cosθ = 6 / (2√2

   3) = 1 / √2 = √2 / 2.

6. 

   6

   Find the angle based on the cosine. You can use the arccos or cos^-1 function on your calculator to

   find the angle θ from a known cos θ value.

   For some results, you may be able to work out the angle based on the unit circle.

   Finding an Angle with Cosine
   In our example, cosθ = √2 / 2. Enter «arccos(√2 / 2)» in your calculator to get the angle. Alternatively, find the angle θ on the unit circle where cosθ = √2 / 2. This is true for θ = ^π/₄ or 45º.
   Putting it all together, the final formula is:
   angle θ = arccosine(( • ) / (|||| ||||))

Advertisement

1. 

   1

   Understand the purpose of this formula. This formula was not derived from existing rules. Instead, it was created as a definition of two vectors’ dot product and the angle between them. However, this decision was not arbitrary. With a look back to basic geometry, we can see why this formula results in intuitive and useful definitions.

   • The examples below use two-dimensional vectors because these are the most intuitive to use. Vectors with three or more components have properties defined with the very similar, general case formula.

2. 

   2

   Review the Law of Cosines. Take an ordinary triangle, with angle θ between sides a and b, and opposite side c. The Law of Cosines states that c² = a² + b² -2abcos(θ). This is derived fairly easily from basic geometry.

3. 

   3

   Connect two vectors to form a triangle. Sketch a pair of 2D vectors on paper, vectors and , with angle θ between them. Draw a third vector between them to make a triangle. In other words, draw vector such that + = . This vector = — .^[3]
   

4. 

   4

   Write the Law of Cosines for this triangle. Insert the length of our «vector triangle» sides into the Law of Cosines:

   • ||(a — b)||² = ||a||² + ||b||² — 2||a|| ||b||cos(θ)

5. 

   5

   Write this using dot products. Remember, a dot product is the magnification of one vector projected onto another. A vector’s dot product with itself doesn’t require any projection, since there is no difference in direction.^[4]
   This means that • = ||a||². Use this fact to rewrite the equation:

6. 

   6

   Rewrite it into the familiar formula. Expand the left side of the formula, then simplify to reach the formula used to find angles.

Advertisement

Advertisement

Video

• If you are working on a computer graphics program, you most likely only care about the direction of the vectors, not their length. Take these steps to simplify the equations and speed up your program:^{[5]
  [6]}
  

• For a quick plug and solve, use this formula for any pair of two-dimensional vectors: cosθ = (u₁ • v₁ + u₂ • v₂) / (√(u₁² • u₂²) • √(v₁² • v₂²)).

Did this summary help you?