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In mathematics, a vector is any object that has a definable length, known every bit magnitude, and direction. Since vectors are not the same equally standard lines or shapes, you'll demand to use some special formulas to detect angles between them.

1. 1

   Write downward the cosine formula. To discover the angle θ betwixt two vectors, start with the formula for finding that angle's cosine. You tin acquire about this formula beneath, or only write it down:^[1]

2. ii

   Identify the vectors. Write downwardly all the information you have concerning the ii vectors. Nosotros'll assume you only have the vector's definition in terms of its dimensional coordinates (too called components). If you already know a vector'due south length (its magnitude), you'll be able to skip some of the steps beneath.

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3. iii

   Calculate the length of each vector. Picture a correct triangle fatigued 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 hands extended to vectors with any number of components.

4. 4

   Calculate the dot product of the two vectors. You accept probably already learned this method of multiplying vectors, as well 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 estimator graphics programs, run into Tips before you lot continue.

   Finding Dot Product Example
   In mathematical terms, ${\displaystyle {\overrightarrow {u}}}$ • ${\displaystyle {\overrightarrow {v}}}$ = u₁v_one + u₂5_two , where u = (u₁, u₂). If your vector has more than two components, only continue to add + u₃v₃ + u₄v₄...
   In our example, ${\displaystyle {\overrightarrow {u}}}$ • ${\displaystyle {\overrightarrow {five}}}$ = u₁v_i + u_twov₂ = (2)(0) + (2)(iii) = 0 + 6 = 6 . This is the dot product of vector ${\displaystyle {\overrightarrow {u}}}$ and ${\displaystyle {\overrightarrow {v}}}$ .

5. 5

   Plug your results into the formula. Retrieve,

   cosθ = ( ${\displaystyle {\overrightarrow {u}}}$ • ${\displaystyle {\overrightarrow {v}}}$ ) / (|| ${\displaystyle {\overrightarrow {u}}}$ || || ${\displaystyle {\overrightarrow {5}}}$ ||).

   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 = half dozen / (2√ii

   iii) = ane / √2 = √2 / ii.

6. 6

   Find the angle based on the cosine. You tin can apply the arccos or cos^-ane function on your calculator to

   find the angle θ from a known cos θ value.

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

   Finding an Angle with Cosine
   In our example, cosθ = √2 / ii. Enter "arccos(√ii / 2)" in your computer to get the bending. Alternatively, find the angle θ on the unit circle where cosθ = √ii / ii. This is true for θ = ^π/₄ or 45º.
   Putting information technology all together, the terminal formula is:
   bending θ = arccosine(( ${\displaystyle {\overrightarrow {u}}}$ • ${\displaystyle {\overrightarrow {v}}}$ ) / (|| ${\displaystyle {\overrightarrow {u}}}$ || || ${\displaystyle {\overrightarrow {5}}}$ ||))

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1. 1

   Understand the purpose of this formula. This formula was not derived from existing rules. Instead, it was created as a definition of 2 vectors' dot product and the bending between them.^[3] Withal, 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 beneath apply two-dimensional vectors because these are the virtually intuitive to use. Vectors with three or more components have properties defined with the very similar, full general case formula.

2. 2

   Review the Law of Cosines. Have an ordinary triangle, with angle θ betwixt 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 2 vectors to form a triangle. Sketch a pair of 2D vectors on newspaper, vectors ${\displaystyle {\overrightarrow {a}}}$ and ${\displaystyle {\overrightarrow {b}}}$ , with angle θ betwixt them. Describe a 3rd vector betwixt them to make a triangle. In other words, draw vector ${\displaystyle {\overrightarrow {c}}}$ such that ${\displaystyle {\overrightarrow {b}}}$ + ${\displaystyle {\overrightarrow {c}}}$ = ${\displaystyle {\overrightarrow {a}}}$ . This vector ${\displaystyle {\overrightarrow {c}}}$ = ${\displaystyle {\overrightarrow {a}}}$ - ${\displaystyle {\overrightarrow {b}}}$ .^[4]

4. 4

   Write the Constabulary 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. v

   Write this using dot products. Remember, a dot product is the magnification of one vector projected onto some other. A vector's dot product with itself doesn't require any projection, since in that location is no difference in direction.^[5] This means that ${\displaystyle {\overrightarrow {a}}}$ • ${\displaystyle {\overrightarrow {a}}}$ = ||a|| ². Use this fact to rewrite the equation:

6. half dozen

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

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Add together New Question

• Question

  If |A + B| = |A| + |B|, and so what is the angle between A and B?

  

  Recollect of the geometric representation of a vector sum. When two vectors are summed they create a new vector past placing the kickoff point of ane vector at the end indicate of the other (write the two vectors on paper). Now, imagine if vectors A and B both where horizontal and added. They would create a vector with the length of their two lengths added! Hence the solution is cypher degrees.

• Question

  How do I find the angle between two vectors if they have the same magnitude?

  

  It depends on their direction. You can't call them vectors without defining their direction.

• Question

  Can you help me solve this problem? "Position vector of the point P and Q relative to the origin O are 2i and 3i+4j respectively. Detect the angles between vector OP and OQ."

  

  An easier way to find the angle between ii vectors is the dot product formula(A.B=|A|x|B|xcos(10)) let vector A exist 2i and vector be 3i+4j. Every bit per your question, Ten is the bending between vectors so: A.B = |A|10|B|10 cos(X) = 2i.(3i+4j) = 3x2 =6 |A|x|B|=|2i|10|3i+4j| = 2 10 5 = x 10 = cos-ane(A.B/|A|ten|B|) Ten = cos-1(vi/ten) = 53.13 deg The angle can be 53.thirteen or 360-53.13 = 306.87.

• Question

  Why can I non use cross products to observe the angles?

  

  Yous can utilize cross products to find the angles, but then you would get the answers in terms of sine.

• Question

  Is at that place any way to discover the angle between vectors other than dot product?

  

  Y'all tin can employ cross product or the cosine formula to determine the angles between the two vectors.

• Question

  How do I discover the angle betwixt perpendicular vectors?

  

  "Perpendicular" ways the bending between the two vectors is 90 degrees. To determine whether the 2 vectors are perpendicular or non, take the cantankerous production of them; if the cross product is equal to zero, the vectors are perpendicular.

• Question

  If 2 or more angles are given with respect to the 10-axis or y-axis, how can I observe the magnitude?

  

  To find the magnitude of more than 2 vectors, instead of using the triangle y'all can use the polygon police to get the answer.

• Question

  How can I summate a unit vector of a given vector?

  

  Presumably, you are asking how to normalize the vector so its magnitude is 1.0. To do that, work out the foursquare root of the sum of the squares of the elements. So, split up each element by this corporeality. What you are doing is scaling the vector so that the sum of the squares equals 1.

• Question

  How tin I find the angle between vectors who brand a dot product of nil?

  

  Siphilisiwe Munzvengi

  Community Answer

  If the dot product is zero, that but means they are perpendicular; therefore, the angle is ninety.

• Question

  How do you discover angle between ii planes defined by say ; 4x-3y+2z and 5x+2y-6z?

  

  To find the angle betwixt two non-parallel planes, you accept to compute the bending betwixt their corresponding normal vectors. By the mode, the examples of airplane equations you gave are not complete.

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• For a quick plug and solve, utilize this formula for whatsoever pair of two-dimensional vectors: cosθ = (u_one • v_i + u₂ • v₂) / (√(u_one ⁱⁱ • u₂ ^two) • √(v₁ ^two • 5₂ ⁱⁱ)).

• If you are working on a calculator graphics program, you most likely merely care near the direction of the vectors, non their length. Take these steps to simplify the equations and speed upward your program:^{[six] [7]}

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Commodity Summary X

one. Summate the length of each vector.
2. Calculate the dot product of the 2 vectors.
3. Calculate the bending between the ii vectors with the cosine formula.
4. Utilise your reckoner's arccos or cos^-ane to find the angle. For specific formulas and example problems, keep reading below!

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