 ### Find the derivative, r'(t), of the vector function. r(t)=e^(t^2)i-j+ln(1+3t)k

The main purpose of this question is to find the derivative of a given vector-valued function. A vector function accepts one or perhaps many variables and yields a vector. Computer graphics, computer vision,  and machine learning algorithms frequently use vector-valued functions. They are especially helpful for determining space curve parametric equations. It is a function possessing two characteristics such as […]

### Find, correct to the nearest degree, the three angles of the triangle with the given vertices. A(1, 0, -1), B(3, -2, 0), C(1, 3, 3).

The main objective of this question is to find the three angles of a triangle given three vertices. The angles can be found using the dot product of the vectors representing the sides of the triangle. A triangle is a polygon with three-sides that is also referred to as a trigon. Every triangle possesses $3$ sides and […]

### Determine whether the given set S is a subspace of the vector space V.

$V=P_5$,  and $S$ is the subset of $P_5$ consisting of the polynomials satisfying $p(1)>p(0)$. $V=R_3$,  and $S$ is the set of vectors $(x_1,x_2,x_3)$ in $V$ satisfying  $x_1-6x_2+x_3=5$. $V=R^n$ and $S$ is a set of solutions to the homogeneous linear system $Ax=0$, where $A$ is a fixed $mtimes n$ matrix. $V=C^2(I)$, and $S$ is the subset […]

### Find the vectors T, N, and B at the given point. r(t)=< t^2,2/3 t^3,t > and point < 4,-16/3,-2 >.

This question aims to find the Tangent, Normal, and Binormal vectors by using the given point and a function. Consider a vector function, $vec{r}(t)$. If $vec{r}'(t)neq 0$ and $vec{r}'(t)$ exist then $vec{r}'(t)$ is called a tangent vector. The line that passes through the point $P$ and is parallel to the tangent vector, $vec{r}'(t)$, is the […]

### Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and area of the triangle PQR.

Take note of the following points:$P(1,0,1) , Q(-2,1,4) , R(7,2,7)$ Find a nonzero vector orthogonal to the plane through the points $P, Q$, and $R$. Find the area of the triangle $PQR$. The purpose of this question is to find an orthogonal vector and the area of a triangle using the vectors $P, Q,$ and […]