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.

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?

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 […]

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,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 $\vec{r}=\vec{0}+\lambda \vec{b}…………….(1)$ […]

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? 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 $ax+by+cz=d$ […]

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 […]

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

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 […]

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