• Prove Hexagon Law of vector Addition.
PROOF:
Hexagon law of vector addition states that , if five vectors are represented by five
consecutive sides of a hexagon in same order then the last side is the resultant in
opposite order.
Let 𝑎⃗ , 𝑏⃗⃗ , 𝑐⃗ , 𝑑⃗ & 𝑒⃗ be the five vectors represented by the sides 𝐴𝐵
̅̅̅̅̅ , 𝐵𝐶
̅̅̅̅ , 𝐶𝐷
̅̅̅̅, ̅̅̅̅ ̅̅̅̅
𝐷𝐸 & 𝐸𝐹
Respectively of hexagon ABCDEF as shown in figure. By Hexagon law of vectors, we
̅̅̅̅ = 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ + 𝑑⃗ + 𝑒⃗
have to prove that 𝐴𝐹
In triangle ∆𝐴𝐵𝐶 , by triangle law of addition
𝐴𝐶 = 𝑎⃗ + 𝑏⃗⃗
̅̅̅̅̅
In triangle ∆𝐴𝐶𝐷 , by triangle law of addition
̅̅̅̅̅ = 𝐴𝐶
𝐴𝐷 ̅̅̅̅̅ + 𝑐⃗
̅̅̅̅̅ = 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗
𝐴𝐷
In triangle ∆𝐴𝐷𝐸 , by triangle law of addition
̅̅̅̅̅ ̅̅̅̅̅ + 𝑑⃗
𝐴𝐸 = 𝐴𝐷
𝐴𝐸 = 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ + 𝑑⃗
̅̅̅̅̅
In triangle ∆𝐴𝐸𝐹 , by triangle law of addition
̅̅̅̅̅
𝐴𝐹 = ̅̅̅̅̅
𝐴𝐸 + 𝑒⃗
𝐴𝐹 = 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ + 𝑑⃗ + 𝑒⃗
̅̅̅̅̅ HENCE PROVED!
PROOF:
Hexagon law of vector addition states that , if five vectors are represented by five
consecutive sides of a hexagon in same order then the last side is the resultant in
opposite order.
Let 𝑎⃗ , 𝑏⃗⃗ , 𝑐⃗ , 𝑑⃗ & 𝑒⃗ be the five vectors represented by the sides 𝐴𝐵
̅̅̅̅̅ , 𝐵𝐶
̅̅̅̅ , 𝐶𝐷
̅̅̅̅, ̅̅̅̅ ̅̅̅̅
𝐷𝐸 & 𝐸𝐹
Respectively of hexagon ABCDEF as shown in figure. By Hexagon law of vectors, we
̅̅̅̅ = 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ + 𝑑⃗ + 𝑒⃗
have to prove that 𝐴𝐹
In triangle ∆𝐴𝐵𝐶 , by triangle law of addition
𝐴𝐶 = 𝑎⃗ + 𝑏⃗⃗
̅̅̅̅̅
In triangle ∆𝐴𝐶𝐷 , by triangle law of addition
̅̅̅̅̅ = 𝐴𝐶
𝐴𝐷 ̅̅̅̅̅ + 𝑐⃗
̅̅̅̅̅ = 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗
𝐴𝐷
In triangle ∆𝐴𝐷𝐸 , by triangle law of addition
̅̅̅̅̅ ̅̅̅̅̅ + 𝑑⃗
𝐴𝐸 = 𝐴𝐷
𝐴𝐸 = 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ + 𝑑⃗
̅̅̅̅̅
In triangle ∆𝐴𝐸𝐹 , by triangle law of addition
̅̅̅̅̅
𝐴𝐹 = ̅̅̅̅̅
𝐴𝐸 + 𝑒⃗
𝐴𝐹 = 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ + 𝑑⃗ + 𝑒⃗
̅̅̅̅̅ HENCE PROVED!
