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Key Highlights
- The vector product of two vectors involves the multiplication of two vectors denoted by a x b.
- Direction of cross product of two quantities is determined by right hand thimb rule.
- Commutative, distributive and anti-commutative properties are some properties of vector product.
- It is used in the field of engineering, physics, mathematics and quantum mechanics.
- We can easily multiply two or more vectors with the help of cross product and dot product method.
The vector product of two vectors is a binary vector operation in which the third vector is perpendicular to the two original vectors. The operation is carried out in three-dimensional Euclidean space and denoted by x.
- The vector product of two vectors is also known as the cross product of two vectors.
- It is used to calculate the magnitude of the resulting vector by using the right-hand rule.
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Key Terms: Vector Product of Two Vectors, Vectors, Scalar, Cross Product, Right-hand Thumb Rule, Magnitude, Properties of Vector Product
What is a Vector
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A vector is an object with magnitude and direction. It is used to locate a point in space with respect to another point. This mathematical quantity can be used to calculate positions, displacements, velocities, and acceleration.
- It can change the proportions and order of the two vectors, but the result will remain the same in the vector space.
- The quantity is denoted by the symbol such as U, V, and W.
- Two vectors are said to be equal if they have the same direction and magnitude.
- It can be represented by a direct line segment that indicates the direction and magnitude of the quantity.
- The vector product of two vectors can be obtained by two methods, namely, dot product and cross product.
In the diagram below, we can see that the direction of the vector is from the tail to the head.
Representation of Vector
Vector Product of Two Vectors
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A vector product of two vectors, also known as cross product or area product, is formed when two real vectors are joined by binary multiplication of two vectors in a three-dimensional space.
- If we assume the two vectors to be a and b, their vector is denoted by a x b.
- The resultant vector is perpendicular to the original vectors.
- A vector product of two vectors is demonstrated using the formula:
a x b = |a||b|sin(θ)n
Here,
- |a| = length or magnitude of vector a
- |b| = length or magnitude of vector b
- θ = angle between both vectors a and b
- n = unit vector perpendicular to both vectors a and b
Magnitude of the Vector Product
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The magnitude of the vector product is determined by the area of the parallelogram formed by the initial two vectors. It is calculated using the square root of the sum of the squares of the vector components.
- Magnitude is a scalar value that cannot be negative.
- It indicates the length of the vector.
- The magnitude of the vector product of two vectors is the product of their lengths.
Consider two vectors: A = Ax + Ay + Az and B = Bx + By + Bz
Then the magnitude of two vectors is given as:
|A| = √(A)x2+(A)y2+(A)z2
|B| = √(B)x2+(B)y2+(B)z2
Hence, the magnitude of the vector product of two vectors is given by the formula,
|A × B| = |A||B||sinθ|
- The fig.1 diagram below shows that θ (i.e. 360°) is the angle between the two vectors a and b.
- In the case of a vector product, we always consider the smaller angle, which is less than 180°.
- As in the fig.2 diagram below, the n represents the area perpendicular to the plane containing the two original vectors.
Fig. 1 Fig. 2
Direction of the Vector Product
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The direction of the vector product of two vectors depends on the chosen orientation of the space. To understand the direction of a vector product, we can follow the convention of the right-hand screw rule.
- Here, one takes a right-handed screw with its head lying on the plane of the vectors a and b.
- With the screw being perpendicular to the plane surface, if one rolls its head in the direction from a to b, then the tip of the screw advances in the direction of c.
- An easier version of this is the right-hand thumb rule.
- In this, one needs to curl up the fingers of our right hand around a line perpendicular to the plane of the vectors a and b.
- If the fingers are curled up in the direction from a to b, then the stretched thumb should point upwards in the direction of c.
- Another way includes aligning the first finger of the right hand with a and the middle finger with b.
- Then, one needs to position these two fingers and the thumb at right angles, with the thumb pointing in the direction of a × b.
Direction of the Vector Product
Properties of Vector Product
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The vector product has interesting properties, which are as follows:
Anti-Commutative Property
While the dot product result of vectors follows the commutative property, however the result is different in case of cross product. The vector product of two vectors is non-commutative.
- Let’s consider a simple instance to understand this property.
- The product of two vectors, a and b is indicated by a x b.
- Now to calculate the product in a different order, we apply the formula b × a = |b| |a| sin(θ) −n.
- In this case, one sees that the direction of b × a is opposite to that of a × b ).
- Mathematically, one can conclude that b × a = −a × b, thus indicating the vector product of two vectors is not commutative.
Distributive Property
The second property of the vector product is that it is distributive over addition, just like the dot product. This means that a × (b + c) = a × b + a × c. The result remains equivalent even when the direction of the vector product is altered. Therefore, (b + c) × a = b × a + c × a.
Zero Vector Property
According to the zero vector property, if there are two vectors, a and b, the vector product of two vectors would be 0 only if a = 0 or b = 0.
Sample Questions
Ques. Answer: (A) When we calculate the product of two vectors, is a x b equal to b x a
(B) What do you mean by the vector product of two vectors ? (2 marks)
Ans. (A) While the magnitude of the two vectors placed upside down will remain the same, their direction would differ. The correct solution for this would be a x b = -b x a.
(B) The vector product of two vectors signifies a vector that is perpendicular to both the original vectors.
Ques. Answer: (A) What are the rules used to determine the direction of a vector product?
(B)Which property of a vector product makes it similar to a scalar product?(2 marks)
Ans. (A) There are two simple rules to identify the direction of a vector product: the right-handed screw rule and the right hand thumb rule.
(B) A common property shared by both the vector and scalar products is that they are distributive over addition.
Ques. What is the dot product? (2 marks)
Ans.The sine of the angle between the two vectors and their magnitudes are combined to form the dot product of vectors. The result of the dot product of two vectors is in the same plane as the two original vectors. The real positive, real negative, or real zero are all valid values for the dot product.
Ques. Two vectors have their scalar magnitude as ∣a∣=5√3 and ∣b∣ = 2, while the angle between the two vectors is 600.Calculate the cross product of two vectors? (2 marks)
Ans. As we know sin60° = √3/2
The vector product of two vectors is given by= 5√3×2×√3/2 = 15
Ques. Find the magnitude of vector a (5, 12)? (2 marks)
Ans. Given Vector a = (5,12)
|a|= √(x2+y2)
|a|= √(52+122)
|a|= √(25 + 144) = √169
Therefore, | a |= 13
Ques. Find the cross product of the given two vectors: \(\vec{X} = 4\hat{i} + 6\hat{j} + 2\hat{k}\) and \(\vec{Y} = \hat{i} + \hat{j} + \hat{k}\) ? (3 marks)
Ans. Given,
\(\vec{X} = 4\hat{i} + 6\hat{j} + 2\hat{k}\)
\(\vec{Y} = \hat{i} + \hat{j} + \hat{k}\)
We have to write the given vectors in determinant form to find the cross product of two vectors. We can find the cross product of two vectors using the determinant form.
\(\vec{X} \times \vec{Y} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 4 & 6 & 2 \\ 1 & 1 & 1 \end{vmatrix}\)
By expanding,
\(\vec{X} \times \vec{Y} = (6-2)\vec{i} - (4-2)\vec{j} + (4-6)\vec{k}\)
Hence,
\(\vec{X} \times \vec{Y} = 4\vec{i} - 2\vec{j} - 2\vec{k}\)
Ques. Find \(\vec{a} \times \vec{b}\) if \(\vec{a} = 5\hat{i} + \hat{k}\) and \(\vec{b} = \hat{i} + 2\hat{j} + \hat{k}\) ? (3 marks)
Ans. Given,
\(\vec{a} = 5\hat{i} + \hat{k}\)
\(\vec{b} = \hat{i} + 2\hat{j} + \hat{k}\)
So,
\(\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 5 & 0 & 1 \\ 1 & 2 & 1 \end{vmatrix}\)
\(=\hat{i}(0-1) - \hat{j}(5-1) + \hat{k}(5-0)\)
\(= -\hat{i}-4\hat{j}+5\hat{k}\)
Ques. Find the cross product of two vectors \(\vec{a}\) = (1, 2, 3) and \(\vec{b}\) = (7, 8, 9)? (3 marks)
Ans. The cross product is given as,
\(\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ 7 & 8 & 9 \end{vmatrix}\)
\(=[(2 \times 9) - (3 \times 8)]\hat{i} - [(1 \times 9) - (3 \times 7)]\hat{j} + [(1 \times 8) - (2 \times 7)]\hat{k}\)
\(= (18-24)\hat{i}- (9-21)\hat{j}+(8-14)\hat{k}\)
\(= -6\hat{i}-12\hat{j}-6\hat{k}\)
Ques. Find the magnitude of vector a (10, 24)? (2 marks)
Ans. Given Vector a = (10,24)
|a|= √(x2+y2)
|a|= √(102+242)
|a|= √(100 + 576) = √676
Therefore, | a |= 26
Ques. Given vector V, having a magnitude of 90 units & inclined at 90°. Break down the given vector into its two components? (3 marks)
Ans. Given, Vector V having magnitude|V| = 90 units and θ = 90°
Horizontal component (Vx) = V cos θ
Vx = 90 cos 90°
Vx = 90 × 0
Vx = 0 units
Now, Vertical component(Vy) = V sin θ
Vy = 90 sin 90°
Vy = 90 × 1
Vy = 90 units
Ques. Given \(\vec{a} = (\hat{i} + 2\hat{j} - 2\hat{k}) \times (-\hat{i} + 4\hat{k})\). Find the magnitude of \(\vec{a}\)? (4 marks)
Ans. The process is as follows:
\(\vec{a} = (\hat{i} + 3\hat{j} - 2\hat{k}) \times (-\hat{i} + 3\hat{k})\)
\(\vec{a} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & -2 \\ -1 & 0 & 4 \end{vmatrix}\)
\(=\hat{i}(8+2) - \hat{j}(4-2) + \hat{k}(0+2)\)
\(= 10\hat{i}-2\hat{j}+2\hat{k}\)
\(|\vec{a}| = \sqrt{10^2 + 2^2 + 2^2}\)
\(=\sqrt{108}\)
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