One of the most interesting topics in Science is Vectors as they are extremely useful in solving many problems related to the concepts. Hence, students are always on the lookout for the formula to find the angle between two vectors.
The vectors are generally drawn using pointed arrows towards a particular direction. The length of the arrow gets to decide the magnitude of the vector. Sometimes, in the problems, you might come across two different vectors and you have to know how to find the angle between two vectors.
When it comes to Vectors, there is a need to find the angle between them if there are two different ones. We are going to discuss that in this section. A vector is basically a particular object with a length that can be defined. This length is called the Magnitude. Vectors also have a direction. It is important to note that vectors are not necessarily the same as shapes and standard lines.
When it comes to finding the angle between two vectors that is deferred by one singular point (shortest angle) at which there is a need to turn the other vector around to a particular position that is co-direction with the other vector.
The formula of the angle between two vectors can be justified only when there are two different vectors that are deferred by one point. That point is known as the shortest angle and it is at this point that there is a need for turning one of these vectors around so that it becomes co-directional to the position of the other vector.
When we are having a discussion on the directional angles of these vectors, we need to use the Vector angle formula with respect to the X-axis that is positive. That makes us focus mostly on the angle that is made between the 2 different vectors and that too in a very standardized position. We can say that a vector has a standard position when the initial point of the vector is present in the Origin of the axis that is (0, 0).
Let us assume that there are two different vectors that are named a and b. Then the dot that is created between the vectors would be denoted as a.b. Now we are going to assume that these two vectors a and b are properly separated using a particular angle.
Now, in order to find out the measurement of the angle, we will have to solve the equation with the help of the angle between two vectors formula that is given as:
The dot product is known as:
a.b = ∣a∣∣b∣cosθ
So, according to the angle between two vectors formula, it will be:
θ = cos−1∣a∣∣b∣a.b
1. Find the Angle Between the Given Two Vectors 3i + 4j – k and 2i – j + k.
a = 3i + 4j – k and
b = 2i – j + k
So, we can say that the dot product is
a.b = (3i + 4j – k).(2i – j + k)
= (3)(2) + (4)(-1) + (-1)(1)
= 6 - 4 - 1
Thus, a.b = 1.
2. What is a Vector?
Ans: A vector is basically a particular object with a length that can be defined. This length is called the Magnitude. Vectors also have a direction. It is important to note that vectors are not necessarily the same as shapes and standard lines.
3. Is it Important to Know the Formula for the Angle Between Two Vectors?
Ans: Yes, it is really important to know the formula for the angle between two equal vectors as it plays a very important role in understanding the concept in the best way.