**Angle Between Two Vectors Using Cross Product :**

Here we are going to see how to find angle between two vectors using cross product.

**Question 1 :**

Find the angle between the vectors 2i vector + j vector − k vector and i vector+ 2j vector + k vector using vector product.

**Solution :**

Angle between two vectors using vector product

θ = sin^{-1} (|a vector x b vector|/|a vector||b vector|)

= i[1+2]-j[2+1]+k[4-1]

a vector x b vector= 3i vector + 3j vector + 3k vector

|a vector x b vector| = √3^{2} + 3^{2 }+ 3^{2 }= 3√3

|a vector| = √2^{2} + 1^{2 }+ 1^{2 }= √6

|b vector| = √1^{2} + 2^{2 }+ 1^{2 }= √6

θ = sin^{-1} (3√3/√6√6)

θ = sin^{-1} (√3/2)

θ = π/3

**Question 2 :**

Let a vector, b vector, c vector be unit vectors such that a ⋅ b = a ⋅ c = 0 and the angle between b vector and c vector is π/3. Prove that a vector = ± (2/√3) (b × c)

**Solution :**

From given information, we have a ⋅ b = a ⋅ c = 0

From this we may decide that a vector is perpendicular to b vector and a vector is perpendicular to c vector.

a vector is perpendicular to both b vector and c vector.So, a vector is proportional to (b x c) vector

a vector = ± λ (b vector x c vector)

|a vector| = ± λ |(b vector x c vector)| ----(1)

|a vector| = ± λ |b||c| sin θ

1 = ± λ sin π/3

λ = 2/√3

By applying the value λ = 2/√3 in (1), we get

a vector = ± (2/√3) (b × c)

Hence it is proved.

**Question 3 :**

For any vector a vector prove that

|a vector × i vector |^{2}+|a vector × j vector|^{2}+|a vector × k vector|^{2}= 2 |a vector|^{2} .

**Solution :**

|a vector × i vector |^{2}+|a vector × j vector|^{2}+|a vector × k vector|^{2}= 2 |a vector|^{2} .

Let a vector = xi vector + yj vector + zk vector, then

a vector x i vector =

= i[0-0] -j[0-z] + k[0-y]

a vector x i vector = zj vector - yk vector

|a vector x i vector| = √z^{2} + y^{2}

|a vector x i vector|^{2} = (√z^{2} + y^{2})^{2}

|a vector x i vector|^{2} = z^{2} + y^{2 } -----(1)

||| ly

a vector x j vector =

= i[0-z] -j[0-0] + k[x-0]

a vector x j vector = -zi vector + xk vector

|a vector x j vector| = √z^{2} + x^{2}

|a vector x j vector|^{2} = (√z^{2} + x^{2})^{2}

|a vector x j vector|^{2} = z^{2} + x^{2} -----(2)

||| ly

a vector x k vector = i[y-0] -j[x-0] + k[0-0]

a vector x k vector = yi vector - xj vector

|a vector x k vector| = √y^{2} + x^{2}

|a vector x k vector|^{2} = (√y^{2} + x^{2})^{2}

|a vector x k vector|^{2} = y^{2} + x^{2} -----(3)

(1) + (2) + (3) ==>

|a vector × i vector |^{2}+|a vector × j vector|^{2}+|a vector × k vector|^{2 }= 2x^{2} + 2y^{2} + 2z^{2}

= 2(x^{2} + y^{2} + z^{2})

= 2|z|^{2}

Hence proved.

After having gone through the stuff given above, we hope that the students would have understood, "Angle Between Two Vectors Using Cross Product"

Apart from the stuff given in "Angle Between Two Vectors Using Cross Product", if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit the following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**