how to find the angle between two vectors

Angle Between Two Vectors

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

Allow us learn more about the bending between ii vectors both in 2D and 3D along with formula, derivation, and examples.

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

What is Angle Between Two Vectors?

The angle between ii vectors is the angle formed at the intersection of their tails. If the vectors are Non joined tail-tail then nosotros have to join them from tail to tail by shifting 1 of the vectors using parallel shifting. Here are some examples to see how to find the bending between two vectors.

The angle between two vectors is the angle between their rails. If their tails are NOT joined, they have to be joined to measure the angle.

Hither, we can run into that when the head of a vector is joined to the tail of another vector, the bending formed is Non the angle between vectors. Instead, one of them should exist shifted either in the same direction or parallel to itself such that the tails of vectors are joined with each other in club to measure the bending.

Angle Betwixt Ii Vectors Formulas

There are two formulas to detect the bending between two vectors: one in terms of dot product and the other in terms of the cross production. Simply the near unremarkably used formula of finding the bending betwixt two vectors involves the dot product (let the states see what is the problem with the cantankerous product in the next section). Let a and b exist 2 vectors and θ be the angle betwixt them. Then hither are the formulas to find the bending betwixt them using both dot production and cantankerous product:

Bending betwixt 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 cantankerous product of a and b. Notation that the cantankerous production formula involves the magnitude in the numerator every bit well whereas the dot production formula doesn't.

The angle between two vectors formulas are given in terms of both dot product and cross product.

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|.

cos θ = (a · b) / (|a| |b|)

θ = cos^-1 [ (a · b) / (|a| |b|) ]

This is is the formula for the bending between ii vectors in terms of the dot product (scalar production).

Angle Between 2 Vectors Using Cross Product

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

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

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

|a × b| = |a| |b| sin θ

Dividing both sides past |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 cantankerous production (vector product).

How to Detect Angle Between Two Vectors?

Let u.s. encounter some examples of finding the angle between two vectors using dot product in both 2D and 3D. Let us too see the ambiguity of using the cantankerous-production formula to find the angle between two vectors.

Angle Between 2 Vectors in 2D

Let us consider two vectors in second say a = <i, -2> and b = <-2, 1>. Let θ exist the bending between them. Let united states find the bending between vectors using both and dot product and cross product and allow u.s.a. encounter what is ambivalence that a cross product tin cause.

Angle Between 2 Vectors in 2D Using Dot Product

Permit us compute the dot production and magnitudes of both vectors.

a · b = <1, -2> ·<-2, 1> = 1(-2) + (-ii)(1) = -ii - 2 = -4.
|a| = √(i)² + (-2)² = √ane + 4 = √5
|b| = √(-2)² + (one)² = √4 + 1 = √v

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

And then θ = cos^-1 (-4 / √v · √5) = cos^-1 (-4/5)

Nosotros can either employ a figurer to evaluate this directly or we can use the formula cos^-1(-x) = 180° - cos^-110 and and then use the calculator (whenever the dot product is negative using the formula cos^-ane(-10) = 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^-i (-4/v) ≈ 143.13°

Bending Between Ii Vectors in 2D Using Cross Production

Allow usa compute the cross production of a and b.

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

Now we observe its magnitude.

|a × b| = √(0)² + (0)² + (-3)² = iii

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

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

If we apply the estimator to calculate this, θ ≈ 36.87 (or) 180 - 36.87 (equally sine is positive in the 2nd quadrant besides). So

θ ≈ 36.87 (or) 143.13°.

Thus, we got ii angles and there is no prove 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 betwixt two vectors.

Bending Betwixt Two Vectors in 3D

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

a · b = <i, 2, 3> ·<3, -two, 1> = i(3) + (-2)(-2) + three(ane) = 3 - iv + iii = 2.
|a| = √(1)² + (two)² + 3² = √1 + 4 +9 = √14
|b| = √(3)² + (-2)² + 1² = √nine + 4 + 1 = √xiv

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

So θ = cos^-1 (ii / √14 · √14) = cos^-1 (ii / xiv) = cos^-1 (1/7) ≈ 81.79°.

Of import Points on Angle Between Two Vectors:

The angle (θ) between two vectors a and b is establish with the formula θ = cos^-ane [ (a · b) / (|a| |b|) ].
The angle betwixt two equal vectors is 0 degrees as θ = cos^-1 [ (a · a) / (|a| |a|) ] = cos^-1 (|a|²/|a|²) = cos^-1i = 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|^two) =cos^-one 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, and so the angle lies betwixt 0° and ninety°;
if a · b is negative, then the angle lies between 90° and 180°.
The angle between each of the ii vectors among the unit vectors i, j, and k is 90°.

Related Topics:

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

Angle Betwixt Two Vectors Examples

Case 1: If θ is the angle between ii vectors a and b such that |a · b| = |a × b|, and so what is θ?

Solution:

It is given that |a · b| = |a × b|.

Past the definition of dot product and cross product,

| |a| |b| cos θ | = | |a| |b| sin θ \(\hat{due north}\)|

Since \(\hat{n}\) is a unit vector, its magnitude is 1.

|a| |b| cos θ = |a| |b| sin θ

cos θ = sin θ

This happens but when θ = 45°.

Answer: The angle between the two vectors when the dot product and cross product are equal is, θ = 45°.
Example 2: Find the angle between vectors a and b if |a| = 1, |b| = ii, and their dot product is a · b = ane.

Solution:

Let u.s.a. assume that the angle between the vectors a and b is θ.

So we have:

θ = cos^-ane [ (a · b) / (|a| |b|) ]

= cos^-ane (1/(i × 2))

= cos^-1 (1/2)

= 60°

Answer: The required angle is lx°.
Case 3: If a and b are two vectors such that |a| = 3, |b| = √2/3, and their cantankerous product is a unit vector, then what is the angle between them?

Solution:

Since the cross production of a and b is a unit vector (given), |a × b| = 1.

Past using the angle between two vectors using cross product formula:

θ = sin^-one [ |a × b| / (|a| |b|) ]

= sin^-one (one/ (iii × √2/3))

= sin^-one (ane/√2)

= 45°

Answer: The required angle is 45°.

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Practice Questions on Bending Between Two Vectors

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FAQs on Angle Between Two Vectors

What is Meant by Bending Betwixt Ii Vectors?

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 practice the parallel shifting of one or both vectors to find the angle between them.

What is Bending Betwixt Two Vectors Formula?

The angle (θ) between 2 vectors a and b can be constitute using the dot product and the cross product. Here are theangle betwixt two vectors formulas:

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

How to Discover Angle Between Two Vectors?

To find the angle between 2 vectors a and b, nosotros tin can use the dot product formula: a · b = |a| |b| cos θ. If we solve this for θ, we go θ = cos^-1 [ (a · b) / (|a| |b|) ].

What is the Angle Between Two Equal Vectors?

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

If the Angle Betwixt Two Vectors is 90 then What is their Dot Production?

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

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

To observe the bending between ii 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 Betwixt Ii Vectors when the Dot Product is 0?

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

Source: https://www.cuemath.com/geometry/angle-between-vectors/

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