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One method of multiplying two or more vectors is to use the dot product. A scalar quantity is the outcome of the dot product of vectors. The dot product is also referred to as a scalar product as a result. It is, mathematically speaking, the product of the corresponding entries of two number sequences. It is composed geometrically of two vectors' Euclidean magnitudes and the sine of the angle between them. The dot product of vectors has numerous uses in astronomy, engineering, mechanics, and geometry. In the sections that follow, let's talk in-depth about the dot product.
Read more: Orthogonal matrix
| Table of Content |
KeyTerms: Dot product, Vectors, Magnitudes, Equivalent property, Scalar multiplication, Angle between two vectors, Projection of vectors, Sine
What is Dot Product?
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Dot product of vectors is the same as the product of the two vectors' magnitudes and the sine of the angle between them. The dot product of two vectors has a result that is in the same plane as the original two vectors. The dot product can be either a real positive number, a real negative number, or a real zero.
Dot product
Dot Product Definition
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A.B stands for the dot product of two different vectors that are non-zero, and its formula is:
a.b = ab cos θ
wherein θ is the angle that is created when a and b meet, and
0 ≤ θ ≤ π
If a = 0 or b = 0, θ won't be specified, and in this instance,
a.b= 0
Dot Product Geometry Definition
According to its geometric definition, the dot product between two given vectors a and b is represented by:
When a.b = |a||b| cos,
Here, is the angle between the vectors a and b, and |a| and |b| are referred to as the magnitudes of the a and b vectors.
A.b = 0 because cos 90 = 0 if the two vectors are orthogonal, that is, if the angle between them is 90.
Since cos 0 = 1, if the two vectors are parallel, then a.b =|a||b|.
Read more:
| Relevant Concepts | ||
|---|---|---|
| Scalar & Vector quantities | Real number | Magnitude of vectors |
| Pair of angles | Scalar triple product | Cross product |
Dot Product Algebra Definition
The dot product of the two products
Given that a = (a1, a2, a3) and b = (b1, b2, b3), respectively, and
a.b = (a1b1 + a2b2 + a3b3),
according to the dot product algebra.
Properties of Dot Product of Two Vectors
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The following are some characteristics of vectors:
Equivalent Property
a .b = b.a
a.b =|a| b|cos θ
a.b =|b||a|cos θ
Discretionary Property
a.(b + c) = a.b + a.c
Inverse Property
a.(rb + c) = r.(a.b) + (a.c)
Property of Scalar Multiplication
(xa). (yb) = xy (a.b)
Indistinctive Property
Because it is forbidden to combine a scalar with a vector as a dot product.
Orthogonal Qualities
Two vectors are only mutually orthogonal when a.b = 0.
Dot Product of Vectors
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The scalar product of two vectors a and b is given as |a||b| cos, where denotes the angle formed by the vectors a and b when viewed in their respective directions.
Dot Product of Vectors
The following can be used to express the scalar product:
a . b = |a| |b| cosθ
where |a| and |b| stand for the vectors' magnitudes, cos stands for the cosine of the angle formed by the two vectors, and a.b stands for the vectors' dot product.
The angle is not defined in the case where any of the vectors is zero, and in this case, a.b is given as zero.
Projection of Vectors
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The projection of a vector an onto a vector b in the direction of vector b determined by |a| cosθ is known as BP.
Similar to this, |b| cosθ represents the projection of vector b on vector a in the direction of vector a.
The expression for the vector projection in the direction of vector b is
BP = \(\frac{a.b}{|b|}\)
⇒ \(\overrightarrow{BP}\) = \(\frac{a.b}{|b|} \times \hat{b}\)
⇒ \(\overrightarrow{BP}\) = \(\frac{a.b}{|b|} . \frac{b}{|b|}\)
⇒ \(\overrightarrow{BP}\) = \(\frac{a.b}{|b|^2} b\)
Similar to how vector an is expressed, vector b is projected in the opposite direction as
BQ = \(\frac{a.b}{|a|}\)
⇒ \(\overrightarrow{BQ}\) = \(\frac{a.b}{|a|} \times \hat{a}\)
⇒ \(\overrightarrow{BQ}\) = \(\frac{a.b}{|a|} . \frac{a}{|a|}\)
⇒ \(\overrightarrow{BQ}\) = \(\frac{a.b}{|a|^2} a\)
As a result, it is clear that the dot product of two vectors is the magnitude of one vector multiplied by the resolved component of the other vector in the first vector's direction.
Dot Product Angle Between Two Vectors
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The cosine of the angle formed by two vectors is used to determine the angle. The sum of the products of the individual components of the two vectors, divided by the product of the magnitude of the two vectors, is the cosine of the angle between the two vectors. The following equation can be used to calculate the angle between two vectors.
Dot Product Angle Between Two Vectors
Things to remember
- A vector that is orthogonal to the two input vectors is produced when two vectors are cross-products.
- The magnitude of the cross product of two vectors is determined by the area of the parallelogram formed by the initial two vectors, and its direction is determined by the right-hand thumb rule.
- A zero vector results from the cross-product of two parallel or linear vectors.
- The cosine of the angle between the two vectors is equal to the sum of the products of the individual components of the two vectors divided by the product of the magnitude of the two vectors.
- The magnitude of one vector is multiplied by the resolved component of the other vector in the first vector's direction to form the dot product of the two vectors.
- The term "BP" refers to the projection of a vector an onto a vector b in the direction of the vector b determined by |a| cosθ.
- 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.
Sample Questions
Ques. Discover the dot product of the variables a= (1, 2, 3), and b= (4, 5, 6). What kind of angle would the vectors make? (3 marks)
Ans. Using the dot products' formula,
a.b = (a1b1 + a2b2 + a3b3)
The dot product can be calculated to be
= 1(4) + 2(−5) + 3(6)
= 4 − 10 + 18
= 12
It can be inferred that the vectors would form an acute angle since a.b is a positive number.
Ques. The formulas for the two vectors A and B are: A = 2i 3j + 7k and B = 4i + 2j 4k. Find the dot product of the two vectors that are given. (2 marks)
Ans. A.B = (2i − 3j +7k) . (−4i + 2j − 4k)
= 2 (−4) + (−3)2 + 7 (−4)
= −8 − 6 − 28
= −42
Ques. Allow two vectors to exist: [6, 2, -1] and [5, -8, 2]. Find the vectors' dot product. (2 marks)
Ans. Given vectors: Let a and b be [6, 2, -1] and [5, -8, 2] respectively.
a.b = (6)(5) + (2)(-8) + (-1)(2)
a.b = 30 – 16 – 2
a.b = 12
Ques. Give two vectors |a|=4 and |b|=2 with = 60 degrees. Find their dot product. (2 marks)
Ans. a.b = |a||b|cos θ
a.b = 4.2 cos 60°
a.b = 4.2 × (1/2)
a.b = 4
Ques. Calculate the dot product of a=(1,2,3) and b=(4,−5,6). Are the angles formed by the vectors acute, right, or obtuse? (4 marks)
Ans. Using the component equation for the three-dimensional vector dot product,
a⋅b=a1b1+a2b2+a3b3,
Calculate the dot product to be
a⋅b=1(4)+2(−5)+3(6)=4−10+18=12.
One can deduce from the geometric definition that the vectors form an acute angle because a-b is positive.
Ques. If a=(6,−1,3), for what value of c is the vector b=(4,c,−2) perpendicular to a? (3 marks)
Ans. Given,
the dot product of a and b to be zero for them to be perpendicular. Since
a⋅b=6(4)−1(c)+3(−2)=24−c−6=18−c,
18c=0, or c=18, must be satisfied by the number c.
By making sure ab=(6,1,3)(4,18,2)=0, you can confirm that the vector b=(4,18,2) is indeed perpendicular to a.
Ques. Calculate the dot product of c=(−4,−9) and d=(−1,2). Are the angles formed by the vectors acute, right, or obtuse? (3 marks)
Ans. Using the component equation for the two-dimensional vector dot product,
a⋅b=a1b1+a2b2,
Calculate the dot product to be
c⋅d=−4(−1)−9(2)=4−18=−14.
Since c⋅d is negative, we can infer from the geometric definition that the vectors form an obtuse angle.
Ques. Determine the dot product of the variables a=(1,2,3) and b=(4,5,6). Are the angles formed by the vectors acute, right, or obtuse? (3 marks)
Ans. Determine the dot product as
a⋅b=1(4)+2(−5)+3(6)=4−10+18=12
using the component formula for the dot product of three-dimensional vectors, a⋅b=a1b1+a2b2+a3b3,
We can deduce from the geometric definition that the vectors form an acute angle because ab is positive.
Ques. Determine the dot product of the values of a = (-4, -9) and b = (-1, 2). What kind of angle will the vectors make? (3 marks)
Ans. To find the dot product of two-dimensional vectors, use the following formula:
a⋅b = a1b1+a2b2,
determine the dot product as
a⋅b = a1b1+a2b2 = -4(-1) – 9(2)
= 4-18
= -14.
Because ab is negative, we can infer that the vectors form an obtuse angle.
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. The magnitude of vector an is 3, the magnitude of vector b is 4, and the angle between a and b is 60 degrees. (2 marks)
What does a.b represent?
Ans. Apply the equation a.b = |a| × |b| cos(θ),
where |a| = 3, |b| = 4, and θ= 60 degrees.
Consequently, a.b = 3 ×4 × cos(60°) = 12 × 0.5 = 6
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