Angle Between Two Vectors Formulas
Angle between two vectors is easily and most commonly found using scalar product of vectors.
Two vector A and B
Dot product of A and B is given by,
[Tex]\vec{A}.\vec{B} [/Tex]= |A| |B| cosθ.
Special Cases
- When angle between vectors are 0 degree.
That is θ = 0°
⇒ |A| |B| cosθ
⇒ |A| |B| cos0°
⇒ |A| |B| [cos0° = 1]
- When angle between vectors are 180 degree.
⇒ |A| |B| cosθ
⇒ |A| |B| cos180°
⇒ – |A| |B| [cos180° = -1]
- When angle between vectors are 90 degree.
⇒ |A| |B| cosθ
⇒ |A| |B| cos90°
⇒ |A| |B| × 0 [cos90° = 0]
⇒ 0
Formula For Angle Between Two Vectors
Cosine of the angle between two vectors is equal to the sum of the product of the individual constituents of the two vectors, divided by the product of the magnitude of the two vectors.
Two Vectors A and B
[Tex]\vec{A}.\vec{B} [/Tex] =| A | | B | cosθ.
cosθ= [Tex]\frac{\vec{A}.\vec{B}}{|A|.|B|} [/Tex]
θ= cos-1 [Tex]\frac{\vec{A}.\vec{B}}{|A|.|B|} [/Tex]
In cartesian Form,
A = Axi + Ayj + Azk
B= Bxi + Byj + Bzk
cos θ = [Tex]\frac{(Ax.Bx+Ay.By+Az.Bz)}{(\sqrt{Ax^2+Ay^2+Az^2}×\sqrt{Bx^2+By^2+Bz^2})} [/Tex]
Properties of Dot product
- Dot product is commutative
[Tex]\vec{A}.\vec{B}=\vec{B}.\vec{A} [/Tex]
- Dot product is Distributive
[Tex]\vec{A}.(\vec{B}+\vec{C})=(\vec{A}.\vec{B}+\vec{A}.\vec{C}) [/Tex]
Angle between two vectors lies between 0 ≤ θ ≤ 180. When the tails or heads of both the vectors coincide, then the angle between vectors is calculated.
Tail Coincide
Head Coincide
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.