How to Find Angle Between Two Vectors? - Formula, Examples (2023)

The angle between two vectors is the angle between their tails. It can be found either by using the dot product (scalar product) or the cross product (vector product). Note that the angle between two vectors always lie between 0° and 180°.

Let us learn more about the angle between two vectors both in 2D and 3D along with formula, derivation, and examples.

1.What is Angle Between Two Vectors?
2.Angle Between Two Vectors Formulas
3.How to Find Angle Between Two Vectors?
4.FAQs on Angle Between Two Vectors

What is Angle Between Two Vectors?

The angle between two vectors is the angle formed at the intersection of their tails. If the vectors are NOT joined tail-tail then we have to join them from tail to tail by shifting one of the vectors using parallel shifting. Here are some examples to see how to find the angle between two vectors.

How to Find Angle Between Two Vectors? - Formula, Examples (1)

Here, we can see that when the head of a vector is joined to the tail of another vector, the angle formed is NOT the angle between vectors. Instead, one of them should be shifted either in the same direction or parallel to itself such that the tails of vectors are joined with each other in order to measure the angle.

Angle Between Two Vectors Formulas

There are two formulas to find the angle between two vectors: one in terms of dot product and the other in terms of the cross product. But the most commonly used formula of finding the angle between two vectors involves the dot product (let us see what is the problem with the cross product in the next section). Let a and b be two vectors and θ be the angle between them. Then here are the formulas to find the angle between them using both dot product and cross product:

  • Angle between two vectors using dot product is, θ = cos-1 [ (a · b) / (|a| |b|) ]
  • Angle between two vectors using cross product is, θ = sin-1 [ |a × b| / (|a| |b|) ]

wherea · b is the dot product and a × b is the cross product of a and b. Note that the cross product formula involves the magnitude in the numerator as well whereas the dot product formula doesn't.

How to Find Angle Between Two Vectors? - Formula, Examples (2)

Angle Between Two Vectors Using Dot Product

By the definition of dot product, a · b = |a| |b| cos θ. Let us solve this for cos θ. Dividing both sides by |a| |b|.

(Video) Finding The Angle Between Two Vectors - Calculus 3

cos θ = (a · b) / (|a| |b|)

θ = cos-1 [ (a · b) / (|a| |b|) ]

This is is the formula for the angle between two vectors in terms of the dot product (scalar product).

Angle Between Two Vectors Using Cross Product

By the definition of cross product, a × b = |a| |b| sin θ \(\hat{n}\). To solve this for θ, let us take magnitude on both sides. Then we get

|a × b| = |a| |b| sin θ |\(\hat{n}\)|.

We know that \(\hat{n}\) is a unit vector and hence its magnitude is 1. So

|a × b| = |a| |b| sin θ

Dividing both sides by |a| |b|.

sin θ = |a × b| / (|a| |b|)

θ = sin-1 [ |a × b| / (|a| |b|) ]

This is is the formula for the angle between two vectors in terms of the cross product (vector product).

(Video) Dot Product : Find Angle Between Two Vectors , Another Example

How to Find Angle Between Two Vectors?

Let us see some examples of finding the angle between two vectors using dot product in both 2D and 3D. Let us also see the ambiguity of using the cross-product formula to find the angle between two vectors.

Angle Between Two Vectors in 2D

Let us consider two vectors in 2D say a = <1, -2> and b = <-2, 1>. Let θ be the angle between them. Let us find the angle between vectors using both and dot product and cross product and let us see what is ambiguity that a cross product can cause.

Angle Between Two Vectors in 2D Using Dot Product

Let us compute the dot product and magnitudes of both vectors.

  • a · b = <1, -2> ·<-2, 1> = 1(-2) + (-2)(1) = -2 - 2 = -4.
  • |a| = √(1)² + (-2)² = √1 + 4 = √5
  • |b| = √(-2)² + (1)² = √4 + 1 = √5

By using the angle between two vectors formula using dot product, θ = cos-1 [ (a · b) / (|a| |b|) ].

Then θ = cos-1 (-4 / √5 · √5) = cos-1 (-4/5)

We can either use a calculator to evaluate this directly or we can use the formula cos-1(-x) = 180° - cos-1x and then use the calculator (whenever the dot product is negative using the formula cos-1(-x) = 180° - cos-1x is very helpful as we know that the angle between two vectors always lies between 0° and 180°). Then we get:

cos-1 (-4/5) ≈ 143.13°

Angle Between Two Vectors in 2D Using Cross Product

Let us compute the cross product of a and b.

(Video) Example:Finding Angle Between Two Vectors

a × b = \(\left|\begin{array}{ccc}
i & j & k \\
1 & -2 & 0 \\
-2 & 1 & 0
\end{array}\right|\) = <0, 0, -3>

Now we find its magnitude.

|a × b| = √(0)² + (0)² + (-3)² = 3

By using the angle between two vectors formula using cross product, θ = sin-1 [ |a × b| / (|a| |b|) ].

Then θ = sin-1 (3 / √5 · √5) = sin-1 (3/5)

If we use the calculator to calculate this, θ ≈ 36.87 (or) 180 - 36.87 (as sine is positive in the second quadrant as well). So

θ ≈ 36.87 (or) 143.13°.

Thus, we got two angles and there is no evidence to choose one of them to be the angle between vectors a and b. Thus, the cross product formula may not be helpful all the times to find the angle between two vectors.

Angle Between Two Vectors in 3D

Let us consider an example to find the angle between two vectors in 3D. Let a = i + 2j + 3k and b = 3i - 2j + k. We will compute the dot product and the magnitudes first:

  • a · b = <1, 2, 3> ·<3, -2, 1> = 1(3) + (-2)(-2) + 3(1) = 3 - 4 + 3 = 2.
  • |a| = √(1)² + (2)² + 3² = √1 + 4 +9 = √14
  • |b| = √(3)² + (-2)² + 1² = √9 + 4 + 1 = √14

We have θ = cos-1 [ (a · b) / (|a| |b|) ].

Then θ = cos-1 (2 / √14 · √14) = cos-1 (2 / 14) = cos-1 (1/7) ≈ 81.79°.

(Video) How to find the angle between two vectors

Important Points on Angle Between Two Vectors:

  • The angle (θ) between two vectors a and b is found with the formula θ = cos-1 [ (a · b) / (|a| |b|) ].
  • The angle between two equal vectors is 0 degrees as θ = cos-1 [ (a · a) / (|a| |a|) ] = cos-1 (|a|2/|a|2) = cos-11 = 0°.
  • The angle between two parallel vectors is 0 degrees as θ = cos-1 [ (a · ka) / (|a| |ka|) ] = cos-1 (k|a|2/k|a|2) =cos-1 1 = 0°.
  • The angle(θ) between two vectors a and b using the cross product is θ = sin-1 [ |a × b| / (|a| |b|) ].
  • For any two vectors a and b, if a · b is positive, then the angle lies between 0° and 90°;
    if a · b is negative, then the angle lies between 90° and 180°.​​​
  • The angle between each of the two vectors among the unit vectors i, j, and k is 90°.

Related Topics:

  • Position Vector
  • Subtracting Two Vectors
  • Handling Vectors Specified in the i-j form
  • Triangle Inequality in Vector

FAQs on Angle Between Two Vectors

What is Meant by Angle Between TwoVectors?

The angle between two vectors is the angle at the intersection of their tails when they are attached tail to tail. If the vectors are not attached tail to tail, then we should do the parallel shifting of one or both vectors to find the angle between them.

What is Angle Between Two Vectors Formula?

The angle (θ) between two vectors a and b can be found using the dot product and the cross product. Here are theangle between two vectors formulas:

  • Using dot product: θ = cos-1 [ (a · b) / (|a| |b|) ]
  • Using cross product: θ = sin-1 [ |a × b| / (|a| |b|) ]

How to Find Angle Between Two Vectors?

To find the angle between two vectors a and b, we can use the dot product formula: a · b = |a| |b| cos θ. If we solve this for θ, we get θ = cos-1 [ (a · b) / (|a| |b|) ].

What is the Angle Between Two Equal Vectors?

The angle between two vectors a and b is found using the formula θ = cos-1 [ (a · b) / (|a| |b|) ]. If the two vectors are equal, then substitute b = a in this formula, then we get θ = cos-1 [ (a · a) / (|a| |a|) ] = cos-1 (|a|2/|a|2) = cos-11 = 0°. So the angle between two equal vectors is 0.

If the Angle Between Two Vectors is 90 then What is their Dot Product?

The dot product of a and b is a · b = |a| |b| cos θ. If the angle θ is 90 degrees, then cos 90° = 0. Then a · b = |a| |b| (0) = 0. So the dot product of two perpendicular vectors is 0.

How to Find the Angle Between Two Vectors in 3D?

To find the angle between two vectors a and b that are in 3D:

  • Compute their dot product a · b.
  • Compute their magnitudes |a| and |b|.
  • Use the formula θ = cos-1 [ (a · b) / (|a| |b|) ].

What is Angle Between Two Vectors when the Dot Product is 0?

The angle between two vectors is given by θ = cos-1 [ (a · b) / (|a| |b|) ]. When the dot product is 0, from the above formula, θ = cos-1 0 = 90°. So when the dot product of two vectors is 0, then they are perpendicular.

(Video) Determining the Angle Between Two Vectors

FAQs

What is the formula for angle between 2 vectors? ›

The formula for the angle between two vectors, a and b is θ=cos-1( ab/|a||b|).

How do you find the angle between V and U? ›

So we can compute the angle θ between u and v using the dot product: θ = arccos ( u · v u v ) .

How can you identify the angle of the vector? ›

Formula for angle between two Vectors

The cosine of the angle between two vectors is equal to the sum of the product of the individual constituents of the two vectors, divided by the product of the magnitude of the two vectors. =| A | | B | cosθ.

What is the angle between a B and a B? ›

The vector (A+ B) will be In plane containing vector A and vector B. Vector (A × B) will be perpendicular to the plane containing vector A and B. Thus, the angle between (A + B) and (A × B) is 90° or rad. Alternate solution- (A + B) · (A × B) = |A + B| |A × 8| cos α

How do you find the direction angle of a vector given two points? ›

Correct answer:

To find the directional vector, subtract the coordinates of the initial point from the coordinates of the terminal point.

What is the angle between two vectors U and V? ›

The angle θ between the vectors u → and v → is θ = cos1(−1) = π. Thus, the vectors u → and v → are parallel. The angle θ between the vectors u → and v → is θ = cos − 1 ( 0 ) = π 2 .

What is the angle between A → B → and B → A →? ›

Thus, the angle between the vectors a → × b → and b → × a → is 180º. Q.

What is the angle between A +B and a B vectors is? ›

So, the angle between (A+B) and (A×B) is 900.

What is the angle between a vector and the resultant of a? ›

Hence, we can find the angle of resultant vector as Φ = t a n - 1 ( Q s i n θ P + Q c o s θ )

What is the angle angle side formula? ›

Therefore, the side angle side formula or the area of the triangle using the SAS formula = 1/2 × a × b × sin c.

What is the formula for direction of vector? ›

We know that the slope of a line that passes through the origin and a point (x, y) is y/x. We also know that if θ is the angle made by this line, then its slope is tan θ, i.e., tan θ = y/x. Hence, θ = tan-1 (y/x). Thus, the direction of a vector (x, y) is found using the formula tan-1.

What is the angle between the vector A 2i 3j and y axis? ›

The angle which the vector A = 2i +3j makes with the y-axis, where i and are unit vectors along x- and y-axes, respectively, is 2.

What is the angle between two vectors P 2i 3j K and i 2j 4k? ›

Let is the angle between two vectors. It can be calculated using the concept of dot product as : So, the angle between A and B is 90 degrees. Hence, this is the required solution.

What is the angle between the vectors a 2i 4j 6k and b 3i j 2k? ›

Answer : The angle between a and b is 60.

How do you find an angle with B and C? ›

To solve the for the angles when we have the lengths of all three sides, we use the law of cosines. The law of cosines states that angle A=cos−1(a2−b2−c2−2bc) , B=cos−1(b2−a2−c2−2ac) , and C=cos−1(c2−a2−b2−2ab) .

Which is the angle opposite to the side B? ›

Hence, in triangle ABC, the side opposite to angle B (right angle) is called the hypotenuse.

What is the angle between the vectors A and B if a B 0? ›

⇒θ=900.

How do you find the vector A and B? ›

The Vector product of two vectors, a and b, is denoted by a × b. Its resultant vector is perpendicular to a and b. Vector products are also called cross products. Cross product of two vectors will give the resultant a vector and calculated using the Right-hand Rule.

What is angle A and B? ›

a and b are adjacent angles. Adjacent angles add up to 180 degrees. (d and c, c and a, d and b, f and e, e and g, h and g, h and f are also adjacent). d and f are interior angles. These add up to 180 degrees (e and c are also interior).

What is the angle between a and the resultant of a vector b and a vector minus b vector? ›

Solution : (a) `R=(A+B)+(A-B)` ltbrLgt `R=2A` <br> The angle between `A` and `2A` is zero, because they are parallel vectors. Step by step solution by experts to help you in doubt clearance & scoring excellent marks in exams.

How do you find Tan theta between two vectors? ›

A Better Formula

tan(θ)=∥u×v∥u∙v.

What is the angle between AXB and BXA? ›

So, a → × b → and b → × a → are vectors of same magnitude but opposite in directions. Thus, the angle between the vectors a → × b → and b → × a → is 180º. Suggest Corrections.

How do you find the direction and angle of a vector? ›

Direction angle of a vector: The direction angle of a vector v is the angle from the positive x -axis to v . The direction angle, θ , of the vector, v=ai+bj v = a i + b j , can be found using the arctangent of the ratio of the vertical component of the vector and the horizontal component of the vector.

How do you find the angle between vectors and Axis? ›

  1. To find angle between a vector and x axis.
  2. 48.To find angle between a vector and x axis.
  3. The angle θ between the vector ¯p=^i+^j+^k and unit vector along x-axis is.
  4. The angle θ between the vector ¯p=^i+^j+^k and unit vector along x-axis is.
  5. A = i + j . What is the angle between the vector and X-axis.

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