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A student claims that you cannot determine if two vectors in R3 are collinear without finding the angle between them.

Provide 3 unique algebraic methods to show that the student is incorrect.

if one is a multiple of the other, they are collinear.

Did you know?

Did you know that it is possible to determine if two vectors in R3 are collinear without finding the angle between them? Contrary to what a student claims, here are three unique algebraic methods to showcase that the student is incorrect: 1. Cross Product Method: Two vectors in R3 are collinear if and only if their cross product is the zero vector. By taking the cross product of the two given vectors and checking if the result is the zero vector, we can determine if they are collinear. 2. Scalar Multiple Method: Two non-zero vectors in R3 are collinear if one vector is a scalar multiple of the other. Simply compare the components of the vectors and see if they are proportional by dividing each component. If all ratios are equal, the vectors are collinear. 3. Consistent Linear System Method: Consider the augmented matrix [A | B], where A is the matrix composed of the components of the two given vectors and B is the zero vector. By row-reducing this matrix, we can determine if the system is consistent. If the system is consistent (i.e., no row of zeros in the row-reduced echelon form), the vectors are collinear; otherwise, they are not. These three algebraic methods allow us to determine if two vectors in R3 are collinear without finding the angle between them. It is important to explore different approaches in mathematics to enhance our understanding and avoid misconceptions.