What is Vector cross product: Definition and 39 Discussions
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in three-dimensional space
R
3
{\displaystyle \mathbb {R} ^{3}}
, and is denoted by the symbol
×
{\displaystyle \times }
. Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product).
If two vectors have the same direction or have the exact opposite direction from one another (i.e., they are not linearly independent), or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths.
The cross product is anticommutative (i.e., a × b = − b × a) and is distributive over addition (i.e., a × (b + c) = a × b + a × c). The space
R
3
{\displaystyle \mathbb {R} ^{3}}
together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.
Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation or "handedness". The product can be generalized in various ways; it can be made independent of orientation by changing the result to a pseudovector, or the exterior product of vectors can be used in arbitrary dimensions with a bivector or 2-form result. Also, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can, in n dimensions, take the product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. (See § Generalizations, below, for other dimensions.)
So I tried to solve it this way:
The equations of the lines in vector form are
$$(x-2)\hat i+(y-3)\hat j+(z-4)\hat k=\lambda (\hat i+\hat j-K\hat k)$$
$$(x-1)\hat i+(y-4)\hat j+(z-5)\hat k=\mu (K\hat i+2\hat j+1\hat k)$$
Since the lines are some real multiple of the vectors,
For coplanarity...
The solution to the question is attached herewith. I approached in the exact same way and got |c| = 2. Then I thought like this:
the angle between a and a×b is 90°, and the angle between c and a×b is 30° (given). So one of the possibilities is, the angle between a and c is 90-30=60° degree. |a|...
Hi everyone,
I'm having problems with task c
In the task, the norm has already been defined, i.e. ##||\vec{c}||=\sqrt{\langle \vec{c}, \vec{c} \rangle }## I therefore first wanted to calculate the scalar product of the cross product, i.e. ##\langle \vec{a} \times \vec{b} , \vec{a} \times...
Hi
If i calculate the vector product of a and b in cartesian coordinates i write it as a determinant with i , j , k in the top row. The 2nd row is the 3 components of a and the 3rd row is the components of b.
Does this work for sphericals or cylindricals eg . can i put er , eθ , eφ in the top...
That may sound really silly, and that may be due to my lack of understanding of the operations itself, but:
if ##|\vec{a}\times\vec{b}|=|\vec{a}|\cdot|\vec{b}|sin\theta##, being ##\theta## the angle between the two vectors, how could ##\vec{b}\times\vec{a}## be different? Wouldn't it be just the...
Writing both ##\vec{U}## and ##\vec{B}## with magnitude in all the three spatial coordinates:
$$
\vec{U}\times \vec{B}=
(U_{x}\cdot \widehat{i}+U_{y}\cdot \widehat{j}+U_{z}\cdot \widehat{k})\times
(B_{x}\cdot \widehat{i}+B_{y}\cdot \widehat{j}+B_{z}\cdot \widehat{k})$$
From this point on, I...
Starting with LHS:
êi εijk Aj (∇xA)k
êi εijk εlmk Aj (d/dxl) Am
(δil δjm - δim δjl) Aj (d/dxl) Am êi
δil δjm Aj (d/dxl) Am êi - δim δjl Aj (d/dxl) Am êi
Aj (d/dxi) Aj êi - Aj (d/dxj) Ai êi
At this point, the LHS should equal the RHS in the problem statement, but I have no clue where...
Hi,
what does it mean to cross multiply two vectors? I couldn't imagine them in real life.
eg Force vector.
Multiplying Force vector to a scalar value means you multiple the 'Strength' of the force,
Dot multiplication of Force with displacement to get work, means you get the work in...
Homework Statement
The direction of vectors A and B are given below for several cases. For each case, state the direction of A X B.
a) A points east, B points south.
b) A points east, B points straight down.
c) A points straight up, B points north.
d) A points straight up, B points straight...
Homework Statement
This is not a homework problem, I am currently reading the Derivation of potential of a charged particle in Electric and Magnetic field from the book Mechanics by Symon (I attached the image of the page), I need to know how to expand the vector cross product
such as...
Reading a book about 3d math, and I am confused as to what happened on this Vector Cross Product problem. I'm thinking there was just an error that wasn't caught.
For the first row, instead of (3)(8)-(-4)(-5) shouldn't it have been (3)(8)-(4)(-5) and had the same displayed result of 44?
And for...
Homework Statement
Vectors A and B both have magnitude M. Joined at the tails, they create a 30' angle. What is A x B in terms of M?
Homework EquationsThe Attempt at a Solution
0? OR M^2? Sqrt(3)M/3?
Homework Statement
Given a×b=-i-j+3k and c×a=2i-3j+k, find a×(a-2b+c)
Homework Equations
Cross product (DONE WITHOUT MATRICES).
The Attempt at a Solution
a[/B]×b=c=-(b×a)is all I'm getting to at this point
Homework Statement
Use the LC symbol to calculate the following: $$\nabla \times \frac{\vec{m} \times \hat{r}}{r^2}$$
Where ##\vec{m}## is just a vector, and ##\hat{r}## is the unit radial vector and ##r## is the length of the radial vector.
Homework Equations
On the Levi Civita symbol...
Homework Statement
Vectors A & B lie in an xy plane. A has a magnitude 7.4 and an angle 142(deg) with respect to the +x direction. B has components (-6.84i, -7.37j)
B) What is the angle between the -y axis and the direction of the Cross product between A and B?
Homework Equations
Cross...
Homework Statement
If a, b, c, d are all vectors contained in the same plane, explain why
(a X b) X (c X d) = <0,0,0>Homework Equations
The Cross Product!
The Attempt at a Solution
I know that since all of the vectors are in the same plane that means that a cross product between any of the...
If e1 and e2 are vectors in ℝ3 show that e1 x e2 = e3, e2 x e3 = e1 and e3 x e1 = e2. I have tried to prove this but I can't get it.
My attempt:
Step 1: [a1, a2, a3] x [b1, b2, b3] = [a2b3-a3b2, a3b1-a1b3, a1b2-a2b1]
Step 2: [b1, b2, b3] x [a2b3-b2a3, a3b1-a1b3, a1b2-a2b1] =...
Homework Statement
From John Taylor's Classical Mechanics:
Show that definition (1.9) of the cross product is equivalent to the elementary deinition that R x S is perpendicular to both R and S, with magnitude rssinθ and direction given by the right hand rule. [Hint: It is a fact (though...
Homework Statement
A lever is orientated along the y direction in a Cartesian coordinate system. The length of the lever is 0.5m and one end of it is at the origin of the coordinate system. A (3i-5j)N force applied to the other end of the lever. Calculate the Torque produced by the force...
1. Homework Statement
Two vectors are given by A = -6 i + 5 j and B = 1 i + 4 j
Find A X B (answer only in terms of i, j, k)
Find the angle between A and B (answer is terms of degrees)
2. Homework Equations
All I was told was that if I set a 3x3 matrix like this:
i j k
-6 5 0
1 4 0...
Homework Statement
Hello,
I have a plate that needs levelling. I can only twist the back 2 feet threaded feet, while the single front foot is fixed, as in the diagram:
As mentioned above I can only TWIST the back two feet (separately or together) causing the plate to tilt around the...
I understand the cross product of vectors to some degree and i can calculate. But i don't really understand the origin of the cross product
What does a vector cross product mean in physical terms. Vector addition is quite easy to understand. I don't think the cross product is 'multiplication...
Homework Statement
C= B|A| + A|B|
D= A|B|-B|A|
C and D are orthogonal
Find a third vector perpendicular to both C and D
Homework Equations
[AxB] = |A||B|sin(theta)
The Attempt at a Solution
I know that to find the answer I need to find the cross product of C and D. I have done...
Homework Statement
\bigtriangledown\times\\(v\times w)= v(\bigtriangledown\cdot w) - w(\bigtriangledown\cdot v)+ (v\cdot\bigtriangledown)w - (w\cdot\bigtriangledown) v
I've tried expanding left side and get
[v1(dw2/dy+dw3/dz)-w1(dv2/dy+dv3/dz)]i +...
Homework Statement
For the vectors in the figure below, with a = 8, b = 7, c = sqrt(113), give the magnitude and direction of the following cross products.
(See attachment for figure/right triangle).
(a) a x b
(b) a x c
(c) b x c
Homework Equations
\vec{A} \cdot...
[b]1. Find axb: Given the magnitude of lal=3 and the magnitude of lbl=2.
[b]2. Since I have the magnitudes, I thought maybe I could use the equation of axb=lal lbl sin theta.
[b]3. I thought since I am trying to find axb that I could use 90 as the angle theta to find axb. I am...
Hello all,
it might be funny! but i am stuck to it! what is the vector cross product formula in spherical and cylindrical coordinates?!
I know for Cartesian coordinate we have that nice looking determinant. but what about the other coordinates. I had looks to all the math books (like...
Is there such a thing as a cross product for R4 vectors? Can you use the permutation symbol to express it in the same way that it can be expressed in R3?
Would the correct way to write it be: e _{i,j,k,l} u _{j} v _{k}?
Homework Statement
Given that A = 2i + 4j, evaluate each of the following. (Hint: This question can be answered without computation.)
(a) What is AxB when B = 8i + 16j?
(b) What is AxB when B = -8i - 16j?
Homework Equations
AxB=(Axi + Ayj) x (Bxi +Byj)
=(AxBx)(i x i)...
A particle position is described by position vector r = 3i + 2j and the force vector i - 2j acts on the object.
1) Find the torque about an axis through the origin and perpendicular to the xy plane. Draw the two vectors to check your torque direction.
I used the right hand rule and found...
This is the question:
Two vectors A and B have magnitude A = 3 and B = 3. Their vector product is A X B = -5k+2i. What is the angle Between A and B.
OK so I'll start with what I do know.
I do know that the cross product is the magnitude of A times magnitude of B times sin theta of B.
I end...
hi, I'm currently doing a mechanics module at Uni. The thing is, I'm not very sure about rules regarding the vector cross product and dot product.
For example, it says in my notes for angular momentum:
"Introducing polar coordinates
\mathbf{r} = r(cos \Phi \mathbf{i} + sin \Phi...
I have learned just about the right hand rule in vector cross product. How is this proved? Can anybody give an example where the cross product plays an important role and where the vector cross product formula is obeyed?
Our professor just told us that the torque due to a force acting on a body...