Prove that vector i,j and k are mutually orthogonal using the dot product.
What is actually meant by mutually orthogonal?
mutually orthogonal=the three of them are orthogonal (or perpendicular) to each other,, [angle between them is 90 degrees]
first recall the formula for the dot product. for any given vectors A and B,
A(dot)B=|A||B|cos(theta)
where |A| and |B| are the magnitude of vectors A and B respectively
*note: magnitude means you get the squareroot of the sum of the squares of each element in the given vector.
example, the vector <1,-2,4> has a magnitude of squareroot of (1^2+(-2)^2+4^2) or squareroot of 21
*note: A(dot)B is the sum of the products of the respective elements (x, y and z elements) of two given vectors [this is scalar]
example, A<2,-1,3> and B<-3, 2, 0>
A(dot)B = 2*-3 + -1*2 + 3*0 = -8
going back to your question,
i<1,0,0> ; |i|=1
j<0,1,0> ; |j|=1
k<0,0,1> ; |k|=1
note that i(dot)j is zero, as well as i(dot)k and j(dot)k,,
therefore in the equation, A(dot)B=|A||B|cos(theta)
cos(theta) is equal to zero (since 0/1=0)
therefore, theta you will get is 90 degrees (for i(dot)j, i(dot)k and j(dot)k), which means they are mutually orthogonal to each other,,
so there,, i'm sorry for long explanation..
When three vectors, i, j, and k, are said to be mutually orthogonal, it means that each pair of vectors is perpendicular to each other. In other words, the dot product of any two vectors will be equal to zero.
To prove that i, j, and k are mutually orthogonal using the dot product, we need to calculate the dot products between each pair of vectors and show that they all equal zero.
Let's start with the dot product of i and j:
i · j = |i||j|cos(θ), where |i| and |j| are the magnitudes of vectors i and j, and θ is the angle between them.
Since both i and j are standard unit vectors with magnitudes of 1, we have:
|i| = |j| = 1.
The angle θ between i and j is 90 degrees, as they are perpendicular to each other.
Plugging these values into the formula, we get:
i · j = 1 * 1 * cos(90) = 1 * 1 * 0 = 0.
Therefore, the dot product of i and j is zero.
Similarly, we can calculate the dot products between other pairs of vectors:
i · k = 1 * 1 * cos(90) = 0,
j · k = 1 * 1 * cos(90) = 0.
Hence, the dot product of i, j, and k with each other is zero for all combinations.
Thus, we have proven that vectors i, j, and k are mutually orthogonal using the dot product.