Sample Problems Angle between Two Vectors Formula
Problem 1: Find the angle between vectors (If they form an equilateral triangle)
- a and b vectors
- b and c vectors
- a and c vectors
Solution:
- a and b vectors
For vector a and b, head of both the vectors coincide with each other, hence angle between a and b vector is same as the angle between two sides of equilateral triangle = 60°.
- b and c vectors:
From the above figure, we see that head or tail of the b and c vector does not coincide with each other.
So, by using the property- A vector remains unchanged if it is transmitted parallel to itself.
Now we see the tail of vectors b and c are coincide with each other, therefore is the same as the exterior angle make with an equilateral triangle = 120°.
- a and c vectors
For vectors a and c, the tail of both the vectors coincide with each other, hence the angle between the a and c vector is the same as the angle between two sides of the equilateral triangle = 60°.
Problem 2: Find angles between vectors if they form an isosceles right-angle triangle.
- a and b vector
- b and c vector
- a and c vectors
Solution:
- a and b vector
From the above figure, we see that head or tail of a and b vector does not coincide with each other. So, by using the property- A vector remains unchanged if it is transmitted parallel to itself.
Now, a and b vectors tails coincide with each other and make an angle the same as the exterior angle of a right angle isosceles triangle = 135°.
- b and c vector
From the above figure, b and c vector head or tails does not coincide with each other. So, by using the property, a vector remains unchanged if it is transmitted parallel to itself.
Now, b and c vectors tails coincide with each other and make an angle the same as the exterior angle of a right angle isosceles triangle = 135°.
- a and c vectors
From the above figure, a and c vector head or tails do not coincide with each other. So, by using the property- A vector remains unchanged if it is transmitted parallel to itself.
Now, a and c vectors tails coincide with each other and make an angle the same as the right angle of isosceles triangle = 90°.
Problem 3: Find angle between the vectors A = i + j + k and vector B = -2i – 2j – 2k.
Solution:
From the formula,
A = Axi + Ayj + Azk
B= Bxi + Byj + Bzk
cosθ= [Tex]\frac{(Ax.Bx+Ay.By+Az.Bz)}{(\sqrt{Ax^2+Ay^2+Az^2}×\sqrt{Bx^2+By^2+Bz^2})} [/Tex]
Here in the Given question,
A= i + j + k
B= -2i -2j -2k
Substituting the values in the formula
⇒ cosθ = [Tex]\frac{(1.(-2)+1.(-2)+1.(-2))}{(\sqrt{1^2+1^2+1^2}×\sqrt{(-2)^2+(-2)^2+(-2)^2})} [/Tex]
⇒ cosθ = [Tex]\frac{(-2-2-2)}{(\sqrt{1+1+1}×\sqrt{4+4+4})} [/Tex]
⇒ cosθ = [Tex]\frac{-6}{(\sqrt{3}×\sqrt{12})} [/Tex]
⇒ cosθ = [Tex]\frac{-6}{(\sqrt{36})} [/Tex]
⇒ cosθ = -6/6
⇒ cosθ= -1
⇒ θ = 180°
Problem 4: Find angle between vector A = 3i + 4j and B = 2i + j
Solution:
A = Axi + Ayj + Azk
B = Bxi + Byj + Bzk
cosθ = [Tex]\frac{(Ax.Bx+Ay.By+Az.Bz)}{(\sqrt{Ax^2+Ay^2+Az^2}×\sqrt{Bx^2+By^2+Bz^2})} [/Tex]
Here Given,
A= 3i + 4j + 0k
B= 2i + j + 0k
Substituting the values in the formula,
⇒ cosθ = [Tex]\frac{(3.2+4.1+0.0)}{(\sqrt{3^2+4^2+0^2}×\sqrt{2^2+1^2+0^2})} [/Tex]
⇒ cosθ = [Tex]\frac{(6+4+0)}{(\sqrt{9+16+0}×\sqrt{4+1+0})} [/Tex]
⇒ cosθ = [Tex]\frac{(10)}{(\sqrt{25}×\sqrt{5})} [/Tex]
⇒ cosθ = [Tex]\frac{(10)}{(\sqrt{125})} [/Tex]
⇒ θ = cos-1 ([Tex]\frac{(10)}{5.(\sqrt{5})}[/Tex])
⇒ θ = cos-1 ([Tex]\frac{2}{(\sqrt{5})}[/Tex])
Problem 5: Find the angle between vector A = i + j and Vector B = j + k.
Solution:
From the formula,
A = Axi + Ayj + Azk
B = Bxi + Byj + Bzk
cosθ = [Tex]\frac{(Ax.Bx+Ay.By+Az.Bz)}{(\sqrt{Ax^2+Ay^2+Az^2}×\sqrt{Bx^2+By^2+Bz^2})}[/Tex]
Here in the Given question,
⇒ A = i + j
⇒ B = j + k
⇒ cosθ = [Tex]\frac{(1.0+1.1+0.1)}{(\sqrt{1^2+1^2+0^2}×\sqrt{0^2+1^2+1^2})}[/Tex]
⇒ cosθ = [Tex]\frac{(1)}{(\sqrt{1+1+0}×\sqrt{0+1+1})}[/Tex]
⇒ cosθ =[Tex]\frac{1}{(\sqrt{2}×\sqrt{2})}[/Tex]
⇒ θ = cos-1 (1/2)
⇒ θ = 60°
Angle between Two Vectors Formula
Angle between two vectors is the angle between their tails and this angle can be easily found using cross product and dot product of vector formulas. Angle between two vectors always lies between 0° and 180°.
In this article we will learn about, angle between two vectors, definition, formulas, and examples in detail.