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eimsori [14]
2 years ago
13

Let u = (1, k) and v = (2, 1). Find k such that The distance between u and v is 1 u and v are orthogonal The angle between u and

v is pi/3
Mathematics
1 answer:
SVETLANKA909090 [29]2 years ago
7 0

Answer:k=1,k=-2,k=8\pm 5\sqrt{3}

Step-by-step explanation:

Given two vectors

u=1\hat{i}+k\hat{j}

v=2\hat{i}+1\hat{j}

\left ( i\right )Distance between them is given by

|u-v|=\sqrt{\left ( 2-1\right )^2+\left ( 1-k\right )^2}=1

squaring both side

1^{2}+\left ( 1-k\right )^2=1

k^2-2k+1=0

\left ( k-1\right )^2=0

k=1

\left ( ii\right )

angle between u and v is 90 i.e. orthogonal

u\dot v=0

\left ( 1\hat{i}+k\hat{j}\right )\dot \left ( 2\hat{i}+1\hat{j}\right )=0

2+k=0

k=-2

\left ( iii\right )

angle between u & v is \frac{\pi }{3}

u\dot v=|u||v|cos\left (\frac{\pi }{3}\right )

|u|=\sqrt{1^2+k^2}

|v|=\sqrt{2^2+1^2}

2+k=\left ( \sqrt{1+k^2}\right )\left ( \sqrt{5}\right )cos\left ( \frac{\pi }{3}\right )

\left ( 4+2\right )^2=\left ( 1+k^2\right )5

k^2-16k-11=0

k=8\pm 5\sqrt{3}

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