Angle Between Vectors
Geometry is one of the topics that many students love, no matter if they like calculations. Making angles using scale and compass gives us a different kind of joy and relaxation in the world of numbers and multiplication tables. But then comes trigonometry to add a bit of complexity and along with trigonometry, you get your first taste of vectors.
Yes, vectors are also a part of Mathematics and geometry. An angle between two vectors is the smallest angle that can be used for one vector to rotate on its axis so that it aligns with the other vector. Two vectors are needed to produce a scalar quantity, which is said to be a real number.
Today, we will be trying to find the angle between the two vectors using trigonometric formulas. We will be doing it in such a way that it will become easier for students to understand.
(Image Will be Updated Soon)
(Two vectors connected via dot making angle theta.)
If you are looking to find an angle between two vectors using a calculator, you might be in for a surprise. Still, there are many websites online that can show you the direct answer, but that's not how you will get marks in your exams. So, we will be helping to solve it.
Angle between Two Vectors Formula
To find the angle between the vectors, we first need to take two vectors in the equation. Let's assume two vectors and name them vector (X) and vector (Y). Now separate these two vectors with angle.
Here, we have now set up the situation to help us find out the angle between the two vectors. To find out the angle, we first need to find out the given vectors' dot product. As a result, vector (X) and vector (Y) = |X| |Y| Cos.
Thus, making the angle between the two vectors given in the formula will be as follows:
\[ \theta = Cos^{-1}\frac{\overrightarrow{x}.\overrightarrow{y}}{|\overrightarrow{x}||\overrightarrow{y}|}\]
In the above equation, we can find the angle between the two vectors.
This was the easy way to find the angle between two vectors. Let us now go through the two common ways to determine this angle, and then we will decide which one to use for our case.
Two Methods to Calculate the Angle between Two Vectors
There are two major formulas that are generally used to determine the angle between two vectors: one is in terms of dot product and the other is in terms of the cross product. However, the most widely used formula to determine the angle between two vectors involves the dot product method. Now, we will see what problem arises when we use the cross-product method. Consider x and y to be two vectors and θ to be the angle between them. The following are the two formulas that can be used to find the angle between them. These formulas use both the dot product and the cross product.
The angle between two vectors can be determined using the dot product as \[ \theta = cos^{-1} [ \frac{x . y}{ \left | x \right | \left | y \right |}\]
The angle between two vectors can be determined using the cross product as \[ \theta =sin^{-1} [ \frac{x \times y}{ \left | x \right | \left | y \right |}\].
Here, x · y is the dot product and x × y is the cross product of x and y. It is to be noted that the cross product formula requires the magnitude of the numerator, while the dot product formula does not.
Note: When it comes to finding out the angle between two equal vectors, you don’t need to solve any equation as the angle will be zero. The main reason behind it is that two equal vectors will have the same direction and magnitude as one another.
Solved Example
Let's try to use the following equation to determine the angle between the two vectors 3i + 4j - k and 2i - j + k.
The first vector is 3i + 4j - k.
The second vector is 2i - j + k.
Now, let's find the dot product of these two.
= (3i + 4j - k ).(2i - j + k).
= (3)(2) + (4)(-1) + (-1)(1)
= (6-4-1)
= -1
Thus, the dot product of the two vectors = 1.
Now, we have to find out the magnitude of the vectors.
For the first one, \[\sqrt{3^{2}+4^{2}+(-1)^{2}}\] = \[\sqrt{26}\] = 5.09
For the second one, \[\sqrt{2^{2}+(-1)^{2}+1^{2}}\] = \[\sqrt{6}\] = 2.45
Now, putting the values in the formula.,
\[\theta=Cos^{-1}\frac{\overrightarrow{x}.\overrightarrow{y}}{|\overrightarrow{x}||\overrightarrow{y}|}\]
= \[Cos^{-1}\frac{1}{(5.09)(2.45)}\]
= \[Cos^{-1}\frac{1}{(12.47)}\]
= \[Cos^{-1} (0.0802)\]
= 85.39o
Conclusion
To summarise, let us go through the major points that we have learned about this topic. The angle between the tails of two vectors is known as the angle between these vectors. There are two ways in which we can find this angle, that is, either by using the dot product (scalar product) or the cross product (vector product). It must be noted that the angle between two vectors will always lie somewhere between 0° and 180°.
FAQs on Angle Between Two Vectors
1. How do you find the angle between two vectors using a formula?
To find the angle (θ) between two vectors, you use the dot product formula. The formula is: cos θ = (a · b) / (|a| |b|). Here, 'a · b' is the dot product of the two vectors, and '|a|' and '|b|' are their respective magnitudes. Once you calculate the value of cos θ, you can find the angle using the inverse cosine function.
2. What is the condition for two vectors to be perpendicular to each other?
Two non-zero vectors are perpendicular (or orthogonal) if the angle between them is 90 degrees (or π/2 radians). The key condition arising from this is that their dot product is zero. So, if a · b = 0, the vectors 'a' and 'b' are perpendicular.
3. What does an angle of 0° or 180° between two vectors indicate?
The angle between two vectors tells you about their relative direction:
- An angle of 0° means the vectors are parallel and point in the exact same direction.
- An angle of 180° means the vectors are anti-parallel, meaning they are parallel but point in opposite directions.
4. What are some real-life examples where we need to find the angle between two vectors?
Finding the angle between vectors is crucial in many real-world applications. For instance, in physics, it's used to calculate the work done by a force, which depends on the angle between the force and displacement vectors. In computer graphics, it helps determine how light reflects off a surface. It's also used in navigation to find the angle between a ship's path and the direction of the wind or current.
5. Why do we generally use the dot product, and not the cross product, to find the angle between vectors?
We use the dot product because it gives a scalar quantity (a single number), which makes it straightforward to solve for the cosine of the angle. The cross product, on the other hand, results in a new vector that is perpendicular to the original two. While the magnitude of this new vector involves the sine of the angle, using the dot product is a more direct and simpler method for finding the angle itself.
6. Is the formula for the angle between vectors different in 2D and 3D space?
No, the fundamental formula is exactly the same for both 2D and 3D space. The principle of using the dot product and magnitudes remains constant. The only difference is in the calculation: for 3D vectors, you include the z-components when calculating the dot product and magnitudes, whereas in 2D, you only work with x and y components.
7. How is the concept of a vector in Mathematics related to its use in Physics?
In Mathematics, a vector is an abstract object defined by its magnitude and direction. Physics takes this abstract concept and applies it to represent real-world physical quantities. For example, quantities like force, velocity, and displacement are vectors because they all have both a magnitude (how much) and a direction. The mathematical rules for vector operations, like finding the angle between them, are the essential tools used to solve complex problems in physics.
8. Can the angle between two vectors be greater than 180 degrees? Explain why.
No, by convention, the angle between two vectors is always taken as the smaller angle formed when they are placed tail-to-tail. This ensures the value is always between 0° and 180° (or 0 and π radians). While there is a larger reflex angle between them, we always use the principal value, which is the shortest turn from one vector to the other.
