Articles of vectors

How to rewrite a vector cross product?

Can this vector cross product $ \vec F = q \vec v \times \vec B $ be rewritten to $ \vec B = \vec F \times q \vec v $ ? Edit: Take the special case, that all three vectors are orthogonal to each other.

Points on two skew lines closest to one another

Given two skew lines defined by 2 points lying on them as $(\vec{x}_1,\vec{x}_2)$ and $(\vec{x}_3,\vec{x}_4)$. What are the vectors for the two points on the corrwsponding lines, distance between which is minimum? That distance is thus but what are the points where it is achieved?

What exactly is a basis in linear algebra?

I have a brief understanding of bases. But I don’t know if it is right or not. So, I just need someone to correct me if it’s not. When we look for the basis of the image of a matrix, we simply remove all the redundant vectors from the matrix, and keep the linearly independent […]

Does linear dependency have anything to do when determining a span?

Q: Does $\{(1,1) , (2,2)\}$ span $\mathbb{R}^2$? A: No, because they are linearly dependent. I agree that it doesn’t span $\mathbb{R}^2$, but from my understanding, linear dependency has nothing to do with that: All that matters is whether you are capable of producing any vector in $\mathbb{R}^2$ by some sort of linear combination of the […]

Prove that the point of intersection of the diagonals of the trapezium lies on the line passing through the mid-points of the parallel sides

Prove,by vector method,that the point of intersection of the diagonals of the trapezium lies on the line passing through the mid-points of the parallel sides. My Attempt: Let the trapezium be $OABC$ and that the O is a origin and the position vectors of $A,B,C$ be $\vec{a},\vec{b},\vec{c}$.Then the equation of $OB$ diagonal is $\vec{r}=\vec{0}+\lambda \vec{b}…………….(1)$ […]

Find unit vector given Roll, Pitch and Yaw

Is it possible to find the unit vector with: Roll € [-90 (banked to right), 90 (banked to left)], Pitch € [-90 (all the way down), 90 (all the way up)] Yaw € [0, 360 (N)] I calculated it without the Roll and it is \begin{pmatrix} cos(Pitch) sin(Yaw)\\ cos(Yaw) cos(Pitch)\\ sin(Pitch) \end{pmatrix}. How should it […]

Why are the coefficients of the equation of a plane the normal vector of a plane?

Why are the coefficients of the equation of a plane the normal vector of a plane? I borrowed the below picture from Pauls Online Calculus 3 notes: http://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfPlanes.aspx And I think the explanation he provides is great, however, I don’t understand how one of the concepts work. If the equation of a plane is $ax+by+cz=d$ […]

How to find the direction vector of a ball falling off an ellipsoid?

A tiny ball is placed in top of an ellipsoid $3x^2+2y^2+z^2=9$ at $(1,1,2)$. Find the three-dimensional vector $\underline u$ in whose direction the ball will start moving after the ball is released. I feel this problem involves usage of gradients but not sure how to tackle it. EDIT the solution shouldn’t use physics knowledge and […]

How to find line parallel to direction vector and passing through a specific point?

I am give the point $(1,0,-3)$ and the vector $2i-4j+5k$ Find the equation of the line parallel to vector and passing through point $(1,0,-3)$ Could one use the fact that the dot product between the line and the vector? Please give me some direction as where to go for this question. I am so lost

Does a complex number multiplication have a geometric representation and why?

When dealing with complex numbers they can be presented as vectors, at least that is stated in my textbook. And the addition operation defined for complex numbers: $$z_1 + z_2 = x_1 + x_2 + i(y_1 + y_2)$$ fully corresponds with the rules for vector addition. But why the multiplication operation does not have a […]