What are Vectors?
Before we get to know the angle between two vectors, let us first understand what a vector is. A vector quantity has a magnitude and a direction as well, unlike a scalar quantity which only has a magnitude. It is denoted by an arrow (→). The length of the arrow represents its magnitude and the direction of the arrow represents the direction of the vector. Suppose, ‘a’ is a vector quantity, then it will be written as \[\overrightarrow{a}\]. [|\overrightarrow{a}|\] denotes the magnitude of a vector. Acceleration, velocity, displacement, force, and momentum are all examples of vector quantity as they have both magnitude and direction.
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Angle Between Two Vectors
Two vectors are said to be equal when their magnitude and direction is the same. However, when the direction of the two vectors is unequal, they will form an angle between them. The angle between the two vectors is denoted by θ.
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Angle Between Two Vectors Using Dot Product
The dot product formula of two vectors ‘\[\overrightarrow{a}\]’ and ‘\[\overrightarrow{b}\]’ is:
\[\overrightarrow{a}\] \[\cdot\] \[\overrightarrow{b}\] = \[|\overrightarrow{a}|\] \[|\overrightarrow{b}|\] cos θ, where \[|\overrightarrow{a}|\] and \[|\overrightarrow{b}|\] are the magnitude of \[\vec{a}\] and \[\vec{b}\] and is the angle between ‘a’ and ‘b’.
The angle (θ) between two vectors can be found using this formula:
\[Cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\]
Or, \[\theta = cos^{-1} \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\]
Now, to find out the formula of the magnitude of the vector using Pythagoras theorem:
\[|\vec{a}| = \sqrt{u_{1}^{2} + u_{2}^{2}}\], where u1 and u2 are the points of \[\vec{a}\] on the x-axis and y-axis respectively in a 2-dimensional graph.
\[|\vec{b}| = \sqrt{v_{1}^{2} + v_{2}^{2}}\], where v1and v2 are the points of \[\vec{b}\] on the x-axis and y-axis respectively in a 2-dimensional graph.
Therefore, the formula becomes:
\[\theta = cos^{-1} \frac{[\vec{a} \cdot \vec{b}}{\sqrt{u_{1}^{2} + u_{2}^{2}} \sqrt{v_{1}^{2} + v_{2}^{2}}]}\]
Now, \[\vec{a} \cdot \vec{b} = u_{1} \cdot v_{1} + u_{2} \cdot v_{2}\]
Hence, the final formula is:
\[\theta = cos^{-1} \frac{[u_{1} \cdot v_{1} + u_{2} \cdot v_{2}}{\sqrt{u_{1}^{2} + u_{2}^{2}} \sqrt{v_{1}^{2} + v_{2}}]}\]
Angle Between Two Vectors Using Cross Product
The formula of the angle between two vectors using the cross product is as follows:
\[\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| sin \theta \widehat{n}\], where,
\[\widehat{n}\] denotes the unit vector that shows the direction of the multiplication of two vectors.
Solved Examples
1. Compute the angle between two vectors using dot product:-
\[\vec{a} = 2 \widehat{i} + 2\widehat{j} + 2\widehat{k}\]
\[\vec{b} = 3\widehat{i} + 3\widehat{j} + 3\widehat{k}\]
Answer: \[\widehat{i}, \widehat{j}\] and \[\widehat{k}\] are called unit vectors.
The dot product of unit vectors are:
\[\widehat{i} \cdot \widehat{i} = 1\]
\[\widehat{j} \cdot \widehat{j} = 1\]
\[\widehat{k} \cdot \widehat{k} = 1\]
\[\widehat{i} \cdot \widehat{j} =0\]
\[\widehat{j} \cdot \widehat{k} = 0\]
\[\widehat{k} \cdot \widehat{i} = 0\]
Therefore, \[\vec{a} \cdot \vec{b} = (2 \widehat{i} + 2\widehat{j} + 2\widehat{k}) \cdot 3 \widehat{i} + 3\widehat{j} + 3\widehat{k}\]
= (2)(3) + (2)(3) + (2)(3)
= 6 + 6 + 6
= 18
Thus, \[\vec{a} \cdot \vec{b} = 18\].
The magnitude of \[\vec{a}\] and \[\vec{b}\] are:
\[|\vec{a} = \sqrt{(2)^{2} + (2)^{2} + (2)^{2}}\]
\[= \sqrt{4 + 4 + 4}\]
\[= \sqrt{12} = 3.46\]
\[|\vec{b} = \sqrt{(3)^{2} + (3)^{2} + (3)^{2}}\]
\[= \sqrt{9 + 9 + 9}\]
\[= \sqrt{27} = 5.19\]
We know that, \[cos \theta = \vec{a} \cdot \vec{b}/|\vec{a}| |\vec{b}|\]
Or, cos θ = 18/ 3.46* 5.19
Or, cos θ = 18/ 17.95
Or, cos θ = 1.002
Or, θ = cos-1 (1.002) where, 1.002 ≈ 1
Or, θ = cos-1(1)
Or, θ = 0°
2. The value of \[|\vec{a}| = 2, |\vec{b}| = 3\] and \[\vec{a} \cdot \vec{b} = 0\]. Compute the angle between \[\vec{a}\] and \[\vec{b}\] .
Answer: Given, \[cos \theta = \vec{a} \cdot \vec{b}/|\vec{a}| |\vec{b}|\]
Here, \[|\vec{a}| = 2, |\vec{b}| = 3\] and \[\vec{a} \cdot \vec{b} = 0\].
Therefore, cos θ = 0/ 2・3
Or, cos θ = 0/6
Or, cos θ = 0
Or, θ = cos-1(0)
Or, θ = 90°
Hence, \[\vec{a}\] and \[\vec{b}\] are perpendicular to each other since the angle between them is 90°.
FAQs on Angle Between Two Vector Products
1. What are the main formulas to find the angle between two vectors?
There are two primary formulas to find the angle, θ, between two vectors a and b:
- Using the Dot Product (most common): The formula is θ = cos⁻¹((a · b) / (|a| |b|)). This method is straightforward because the dot product gives a scalar value.
- Using the Cross Product: The magnitude of the cross product is given by |a × b| = |a| |b| sin(θ). You can rearrange this to find sin(θ), but it's generally less direct for finding the angle alone.
2. What are some fundamental types of vectors I should know?
Understanding these basic vector types is essential:
- Zero Vector: A vector with a magnitude of zero and no defined direction.
- Unit Vector: A vector with a magnitude of exactly 1. It is used to specify a direction.
- Collinear Vectors: Two or more vectors that are parallel to the same line, regardless of their magnitudes and directions.
- Equal Vectors: Vectors that have the same magnitude and the same direction.
- Negative of a Vector: A vector that has the same magnitude as the original vector but points in the opposite direction.
3. How is the angle between two lines in vector form calculated?
To find the angle between two lines given by the vector equations r₁ = a₁ + λb₁ and r₂ = a₂ + μb₂, you don't use the full equations. Instead, you find the angle between their direction vectors, b₁ and b₂. The formula is the same as for any two vectors: θ = cos⁻¹((b₁ · b₂) / (|b₁| |b₂|)).
4. Where is the concept of the angle between vectors used in real life?
This concept is crucial in many fields, especially physics and engineering. For example:
- In Physics, it's used to calculate Work Done, which is the dot product of the Force and Displacement vectors (W = F · d = Fd cosθ).
- In Computer Graphics, it helps determine how light reflects off a surface by calculating the angle between the light source, the surface normal, and the camera.
- In Navigation, it's used to determine the relative direction between a ship's course and the wind or ocean current.
5. What does an angle of 0°, 90°, or 180° between two vectors imply?
The angle tells you about the orientation of the vectors relative to each other:
- 0° Angle: The vectors are parallel and point in the same direction. Their dot product is maximum.
- 90° Angle: The vectors are orthogonal (perpendicular). Their dot product is zero, which is a key test for perpendicularity.
- 180° Angle: The vectors are anti-parallel, meaning they are parallel but point in opposite directions.
6. Why is the dot product usually preferred over the cross product for finding an angle?
The dot product is preferred because it is simpler and more direct. The dot product (a · b) results in a scalar (a single number), which can be directly plugged into the cos⁻¹ formula. In contrast, the cross product (a × b) results in a new vector that is perpendicular to both a and b. To find the angle, you would first need to calculate this new vector and then find its magnitude, which is an extra, often unnecessary, step.
7. What is the angle between the vector products A × B and B × A?
The angle between the vector products A × B and B × A is always 180 degrees. This is because the cross product is anti-commutative, which means A × B = - (B × A). The negative sign indicates that the two resulting vectors have the exact same magnitude but point in precisely opposite directions, making them anti-parallel.
