Discoverand Master Vectors Solutions

Vectors u and v are orthogonal. If $u=⟨3,1+b⟩$ and $v=⟨5,1-b⟩$ find all possible values for b.

Identify the surface with the given vector equation. $r(s,t)=$$(s,t,t^2-s^2)$
eliptic cylinder
circular paraboloid
hyperbolic paraboloid
plane
circular cylinder

Let ${\displaystyle A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}$ and 'k' be the scalar. Find the formula that relates 'detKA' to 'K' and 'detA''

The position vector ${\displaystyle r\left(t\right)=⟨\ln t,\frac{1}{t^2},t^4⟩}$ describes the path of an object moving in space.
(a) Find the velocity vector, speed, and acceleration vector of the object.
(b) Evaluate the velocity vector and acceleration vector of the object at the given value of ${\displaystyle t=\sqrt{3}}$

use the Laplace transform to solve the given initial-value problem.
$y"+y=f(t)$
$y(0)=0,y^{\prime }(0)=1$ where
${\displaystyle f\left(t\right)=\{\begin{array}{cc}1& 0\le t<\frac{\pi }{2}\\ 0& t\ge \frac{\pi }{2}\end{array}}$

Show that ${\displaystyle C^{\ast }=R_{+}^{\ast }\times T,\text{where}{C}^{\ast }}$ is the multiplicative group of non-zero complex numbers, T is the group of complex numbers of modulus equal to 1, ${\displaystyle R_{+}^{\ast }}$ is the multiplicative group of positive real numbers.

Let U,V be subspaces of Rn. Suppose that $U\mathrm{\perp }V.$ Prove that $\{u,v\}$ is linearly independent for any nonzero vectors $u\in U,v\in V.$

Let u,$v_1$ and $v_2$ be vectors in $R^3$, and let $c_1$ and $c_2$ be scalars. If u is orthogonal to both $v_1$ and $v_2$, prove that u is orthogonal to the vector $c_1{v}_1+c_2{v}_2$.

Let U and W be vector spaces over a field K. Let V be the set of ordered pairs (u,w) where $u\in U$ and $w\in W$. Show that V is a vector space over K with addition in V and scalar multiplication on V defined by
$(u,w)+(u,w)=(u+u,w+w)andk(u,w)=(ku,kw)$
(This space V is called the external direct product of U and W.)

Determine the area under the standard normal curve that lies between
(a) Upper Z equals -2.03 and Upper Z equals 2.03​, 
​(b) Upper Z equals -1.56 and Upper Z equals 0​, and 
​(c) Upper Z equals -1.51 and Upper Z equals 0.68.