Q.2 - Calculus

Question

Examples: Construct examples of the thing(s) described in

the following. Try to find examples that are different than

any in the reading.

(a) Two nonzero vectors in $R^3$ whose dot product is

zero.

(b) A function y = f(x) that is orthogonal to the function

y = sin x at x = 0.

(c) A vector $v\ne i$ such that proj v = i.

Step-by-Step Solution

Answer

(a) The dot product is zero.

(b) The function that is orthogonal to y=sinx is y=x.

(c) The vector is <1,0,0>..

Step 1. Finding the Dot product of Vectors

Consider the vectors $u=<2,0,3>$ and $v=<-3,7,2>$.

 Now,

$$\begin{gathered} u.v=2(-3)+0\left(7\right)+3\left(2\right) \\ \Rightarrow u.v=-6+0+6 \\ \Rightarrow u.v=0 \end{gathered}$$

Step 2. Finding the orthogonal function

Consider the function y=sinx.

We need to find the function that is orthogonal to y=sinx.

Therefore, consider the function, y=x.

Now,

$$\begin{gathered} <x,\sin x>=\int x\sin xdx \\ =-x\cos x+\sin x \\ =0 (at x=0) \end{gathered}$$

Therefore, the function that is orthogonal to y=sinx is y=x at x=0.

Step 3. Finding proj v.

We know,

$pro{j}_{i}v=\frac{i.v}{{\left|\left|i\right|\right|}^2}i$

Now,

$$\begin{gathered} i.v=1\left(1\right)+0\left(2\right)+0\left(3\right) \\ =1+0+0 \\ =1 \end{gathered}$$

Now, again,

$$\begin{gathered} i=<1,0,0> \\ \left|\left|i\right|\right|=\sqrt{1^2+0^2+0^2} \\ \left|\left|i\right|\right|==1 \end{gathered}$$

Therefore,

$$\begin{gathered} pro{j}_{i}v=\frac{1}{{\left(\sqrt{1}\right)}^2}<1,0,0> \\ =<1,0,0> \\ =i \end{gathered}$$

Hence, $pro{j}_{i}v=i$