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Find two unit vectos that are orthogonal to both [0,1,2] and [1,-2,3]

User BaseZen
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1 Answer

3 votes

Answer:

Let the vectors be

a = [0, 1, 2] and

b = [1, -2, 3]

( 1 ) The cross product of a and b (a x b) is the vector that is perpendicular (orthogonal) to a and b.

Let the cross product be another vector c.

To find the cross product (c) of a and b, we have


\left[\begin{array}{ccc}i&j&k\\0&1&2\\1&-2&3\end{array}\right]

c = i(3 + 4) - j(0 - 2) + k(0 - 1)

c = 7i + 2j - k

c = [7, 2, -1]

( 2 ) Convert the orthogonal vector (c) to a unit vector using the formula:

c / | c |

Where | c | = √ (7)² + (2)² + (-1)² = 3√6

Therefore, the unit vector is


([7,2,-1])/(3√(6) )

or

[
(7)/(3√(6) ) ,
(2)/(3√(6) ) ,
(-1)/(3√(6) ) ]

The other unit vector which is also orthogonal to a and b is calculated by multiplying the first unit vector by -1. The result is as follows:

[
(-7)/(3√(6) ) ,
(-2)/(3√(6) ) ,
(1)/(3√(6) ) ]

In conclusion, the two unit vectors are;

[
(7)/(3√(6) ) ,
(2)/(3√(6) ) ,
(-1)/(3√(6) ) ]

and

[
(-7)/(3√(6) ) ,
(-2)/(3√(6) ) ,
(1)/(3√(6) ) ]

Hope this helps!

User Marcel Klehr
by
8.1k points

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