# How To Solve Vector Problems

Visit Stack Exchange A plane is headed due south with an airspeed of 192 mph.A wind with a bearing of 78 degrees is blowing at 23 mph.

Also, determine those vectors $\mathbf\in \R^2$ such that $\mathbf^A\mathbf=0$. Read solution Problem 1 Let $W$ be the subset of the $-dimensional vector space $\R^3$ defined by \[W=\left\.\] (a) Which of the following vectors are in the subset $W$? \[(1) \begin 0 \ 0 \ 0 \end \qquad(2) \begin 1 \ 2 \ 2 \end \qquad(3)\begin 3 \ 0 \ 0 \end \qquad(4) \begin 0 \ 0 \end \qquad(5) \begin 1 & 2 & 4 \ 1 &2 &4 \end \qquad(6) \begin 1 \ -1 \ -2 \end.\] (b) Determine whether $W$ is a subspace of $\R^3$ or not.(c) If $\mathbf_1, \mathbf_2$ are linearly independent vectors and $A$ is nonsingular, then show that the vectors $A\mathbf_1, A\mathbf_2$ are also linearly independent.Read solution (a) For what value(s) of $a$ is the following set $S$ linearly dependent?Find the groundspeed and resulting bearing of the plane.I tried doing it this way as I was not given an angle for the airspeed of the plane for the first one, I don't know if the component form of the wind is correct or not, I'm confused on the idea of converting the bearing of an angle.Since the plane is moving South with respect to the air, the new components along ground are going to be 2-23\cos$mph towards South and \sin$mph towards East.Let $A=\begin 1 & 0 & 3 & -2 \ 0 &3 & 1 & 1 \ 1 & 3 & 4 & -1 \end$.Read solution In this problem, we use the following vectors in $\R^2$.\[\mathbf=\begin 1 \ 0 \end, \mathbf=\begin 1 \ 1 \end, \mathbf=\begin 2 \ 3 \end, \mathbf=\begin 3 \ 2 \end, \mathbf=\begin 0 \ 0 \end, \mathbf=\begin 5 \ 6 \end.\] For each set $S$, determine whether $\Span(S)=\R^2$.(a) Prove that $\mathbf^\trans \mathbf = \mathbf^\trans \mathbf$.(b) Provide an example to show that $\mathbf \mathbf^\trans$ is not always equal to $\mathbf \mathbf^\trans$. (d) $\mathbf^\trans \mathbf \mathbf^\trans B A^\trans \mathbf$.

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