Answer to Question #245546 in Analytic Geometry for moe

Question #245546

Consider the vectors a=(k,1+k,k(1+k))\mathbf{a}=(-k, 1+k, k(1+k)) and b=(1+k,k(1+k),k)\mathbf{b}=(1+k, k(1+k), -k) where k is some constant. Which of the following is/are correct?



  1. a and b are perpendicular vectors for any k
  2. a∣=∣b∣.
  3. a and b are unit vectors for k=1
  4. The values of k for which ∣a∣=1 are 0 and -1
  5. The two vectors above are parallel.
1
Expert's answer
2021-10-21T14:55:24-0400

1.

If two vectors are perpendicular then A.B=0\overrightarrow{A}.\overrightarrow{B}=0

[ki^+(i+k)j^+9k+k2)k^].[(1+k)i^+(k+k2)j^kk^=0[-k\widehat{i}+(i+k)\widehat{j}+9k+k^2)\widehat{k}].[(1+k)\widehat{i}+(k+k^2)\widehat{j}-k\widehat{k}=0

k(1+k)+(1+k)(k+k2)+(k+k2)(k)=0-k(1+k)+(1+k)(k+k^2)+(k+k^2)(-k)=0

kk2+k+k2+k2+k3k2k3=00=0-k-k^2+k+k^2+k^2+k^3-k^2-k^3=0\\0=0

option 1 is correct


2.

A=(k)2+(1+k)2+(k+k2)|\overrightarrow{A}|=\sqrt{(-k)^2+(1+k)^2+(k+k^2)}


=k2+1+k2+2k+k2+k4+2k3\\=\sqrt{k^2+1+k^2+2k+k^2+k^4+2k^3}


=k4+2k3+3k2+2k+1=\sqrt{k^4+2k^3+3k^2+2k+1}


B=(1+k)2+(k+k2)+(k)2\\|\overrightarrow{B}|=\sqrt{(1+k)^2+(k+k^2)+(-k)^2}


=1+k2+2k+k2+k4+2k3+k2\\=\sqrt{1+k^2+2k+k^2+k^4+2k^3+k^2}


k4+2k3+3k2+2k+1\\\sqrt{k^4+2k^3+3k^2+2k+1}


A=B\\|\overrightarrow{A}|=|\overrightarrow{B}|


option 2 is correct


3.

if A&B\overrightarrow{A} \& \overrightarrow{B} is a unit vector


A=1, B=1|\overrightarrow{A}|=1,\space |\overrightarrow{B}|=1


k4+2k3+3k2+2k+1=1k(k3+2k2+3k+2)=0................(1)for k=1k^4+2k^3+3k^2+2k+1=1\\k(k^3+2k^2+3k+2)=0................(1)\\for\space k=1

equation (1) not satisfied so option (3) is not correct


4.

A=1|\overrightarrow{A}|=1

by equation (1)

k((k3+2k2+3k+2)=0k=0k3+2k2+3k+2=0put k=11+23+2=00=0k((k^3+2k^2+3k+2)=0\\k=0\\k^3+2k^2+3k+2=0\\put\space k=-1\\-1+2-3+2=0\\0=0


option 4 is correct


5.

vector is parallel if

k=d(1+k) & 1+k=d(k(1+k)k1+k=d, 1k=d & k(1+k)=d(k), d=(1+k)-k=d(1+k)\space \&\space 1+k=d(k(1+k)\\\frac{-k}{1+k}=d,\space \frac{1}{k}=d\space \&\space k(1+k)=d(-k),\space d=(-1+k)

Therefore the vectors are not parallel


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