- Books Name
- Mathmatics Book Based on NCERT
- Publication
- KRISHNA PUBLICATIONS
- Course
- CBSE Class 12
- Subject
- Mathmatics
Vector (or Cross) Product of Two Vectors
Vectors can be multiplied in two ways, a scalar product where the result is a scalar and cross or vector product where is the result is a vector. In this article, we will look at the cross or vector product of two vectors.
Explanation
We have already studied the three-dimensional right-handed rectangular coordinate system. As shown in the figure below, when the positive x-axis is rotated counter-clockwise into the positive y-axis, then a right-handed standard screw moves in the direction of the positive z-axis.
As can be seen above, in a three-dimensional right-handed rectangular coordinate system, the thumb of the right-hand points in the direction of the positive z-axis when the fingers are curled from the positive x-axis towards the positive y-axis.
Definition
The cross or vector product of two non-zero vectors a⃗ and b⃗ , is
a⃗ x b⃗ = |a⃗ | |b⃗ | sinθn^
Where θ is the angle between a⃗ and b⃗ , 0 ≤ θ ≤ π. Also, n^ is a unit vector perpendicular to both a⃗ and b⃗ such that a⃗ , b⃗ , and n^ form a right-handed system as shown below.
As can be seen above, when the system is rotated from a⃗ to b⃗ , it moves in the direction of n^. Also, if either a⃗ = 0 or b⃗ = 0, then θ is not defined and we can say,
a⃗ x b⃗ = 0⃗
Important Observations
- a⃗ x b⃗ is a vector.
- If a⃗ and b⃗ are two non-zero vectors, then a⃗ x b⃗ = 0, if and only if a⃗ and b⃗ are parallel (or collinear) to each other, i.e.
a⃗ x b⃗ = 0 ⇔ a⃗ ∥ b⃗
Hence, a⃗ x a⃗ = 0 and a⃗ x (−a)→ = 0. This is because in the first case θ = 0. Also, in the second case θ = π, giving the value of sinθ = 0.
- If θ = π2, then a⃗ x b⃗ = |a⃗ | |b⃗ |
- Considering observations 2 and 3 above, for mutually perpendicular vectors i⃗ , j⃗ , and k⃗ , we have
- The angle between the two vectors a⃗ and b⃗ is,
sinθ = |a⃗ ×b⃗ ||a⃗ ||b⃗ |
- A cross or vector product is not commutative. We know this because a⃗ x b⃗ = −b→ x a⃗ . Now, we know that,
a⃗ x b⃗ = |a⃗ | |b⃗ | sinθn^.
Where a⃗ , b⃗ , and n^ form a right-handed system. Or, θ is traversed from a⃗ to b⃗ . On the other hand,
b⃗ x a⃗ = |b⃗ | |a⃗ | sinθn1^.
Where b⃗ , a⃗ , and n1^ form a right-handed system. Or, θ is traversed from b⃗ to a⃗ . So, if a⃗ and b⃗ lie on a plane of paper, then n^ and n1^ are both perpendicular to the plane of the paper. However, n^ is directed above the paper and n1^ is directed below it. Or, n^ = – n1^. Hence,
a⃗ x b⃗ = |a⃗ | |b⃗ | sinθn^ = – |a⃗ ||b⃗ |sinθ n1^
= – b⃗ x a⃗
- From the observations 4 and 6 above, we have
j^ x i^ = – k^
k^ x j^ = – i^
i^ x k^ = – j^
- If a⃗ and b⃗ represent the two sides of a triangle, then its area is |a⃗ x b⃗ |. To understand this, look at the figure given below.
By the definition of the area of a triangle, we have area of ΔABC = (AB).(CD). We know that, AB = |b⃗ | and CD = |a⃗ |sinθ. Therefore,
- If a⃗ and b⃗ represent the two adjacent sides of a parallelogram, then its area is |a⃗ x b⃗ |. To understand this, look at the figure given below.
By the definition of the area of a parallelogram, we have area of parallelogram ABCD = (AB).(DE). We know that, AB = |b⃗ | and DE = |a⃗ |sinθ. Therefore,
Area of parallelogram ABCD = |a⃗ ||b⃗ |sinθ = |a⃗ x b⃗ |
Property: Distributivity of a cross or vector product over addition
If a⃗ , b⃗ , and c⃗ are any three vectors and λ is a scalar, then
- a⃗ x (b⃗ + c⃗ ) = a⃗ x b⃗ + a⃗ x c⃗
- λ(a⃗ x b⃗ ) = (λa⃗ ) x b⃗ = a⃗ x (λb⃗ )
Question 1: Find the area of the parallelogram whose adjacent sides are determined by the following vectors,
- a⃗ = i^ – j^ + 3k⃗ and
- b⃗ = 2i^ – 7j^ + k⃗ .
Answer : We know that if a⃗ and b⃗ represent the two adjacent sides of a parallelogram, then its area is |a⃗ x b⃗ |. Also,
Substituting the values of a1,a2,a3,b1,b2,and b3, we get
Solving the determinant, we get
- a⃗ x b⃗ = {[(-1) x 1)] – [(-7) x 3]} – {[1 x 1)] – [2 x 3]} + {[1 x (-7))] – [2 x (-1)]}
= 20i^ + 5j^ – 5k^.
Also, the magnitude of a⃗ x b⃗ is,
- |a⃗ x b⃗ | =√ [20+5+(−5)]= √450 = √25×9×2 = √152.
Therefore, the area of the parallelogram is √152.
Question : Explain the characteristics of vector product?
Answer: The characteristics of vector product are as follows:
- Vector product two vectors always happen to be a vector.
- Vector product of two vectors happens to be noncommutative.
- Vector product is in accordance with the distributive law of multiplication.
- If a • b = 0 and a ≠ o, b ≠ o, then the two vectors shall be parallel to each other.