For the following exercises, determine whether the statement...

For the following exercises, determine whether the statement is true or false. Justify the answer with a proof or a counterexample.
For vectors a and $\mathbf{b}$ and any given scalar $c,$ $c(\mathbf{a} \times \mathbf{b})=(c \mathbf{a}) \times \mathbf{b}$.

Key Concepts

Bilinearity

Bilinearity refers to the property of an operation being linear in each of its arguments. For the cross product, this means that the operation satisfies distributivity over vector addition and compatibility with scalar multiplication in each slot independently.

Scalar Multiplication

Scalar multiplication in the context of vector operations involves multiplying each component of a vector by a scalar. When combined with bilinear operations like the cross product, scalar multiplication can be 'pulled out' of the operation, meaning that multiplying one of the vectors by a scalar is equivalent to multiplying the result of the cross product by that scalar.

Cross Product

The cross product is an operation on two vectors in three-dimensional space that results in a new vector that is perpendicular to both operands. Its magnitude is equal to the area of the parallelogram formed by the original vectors, and it depends on the sine of the angle between them.

Transcript

00:01 Alright, so we have two vectors a and the b and we do the cross product between these two.

00:16 Is it true that if c is some real number, is that equal to just multiplying the c by a and then the cross product that would be? so is that equal well let's see so we have a equals to some a 1 for the x coordinate a 2 and a 3 for the c coordinate and b equals to or be 1 b1 b1 b sub 2 be 3 what is in that product well that product is going to be equal to just computing the determinant of the matrix of here i j k so this a coordinated representation is the same as same for the i component you have a 1 so a 1 times i plus a 2 times j and plus a three times k so these are those are the corresponding unit vectors i j and k for the coordinates x y and c x corresponding to y is j is j so the cross product is going to be the same as computing the determinant of this matrix a1, a2, a3, and then you put below here b1, b2, b3, so that all the cross product here is going to be turning on this i times, this times that, minus that, and that.

02:59 So it's i times a2 times b3, a2, b2, b3, a2, b3.

03:14 Minus a 3 times b2 now for the j component is going to be equal to for the j we have so it's this one so remove that so it's a1 with v 3 minus b1 with a 2 so j so j times a 1 b3 minus a 3 b1 now for the k component it's equal to let's do blue um for the k so you remove that and that so you have a1 times b2 minus b1 times a 2 so that is a 1 times a 2 so that is a 1 times b2 so that is a 1 minus 8 2 times p1 now well what happens if i multiply all that stuff by c well well it all gets multiply by c so it's gonna be a c in here let's see them see in there so this part that is the left hand side of this equation now so what happens if we the right with the right hand side of that equation so if you do the cross product of c a cross b as you see or c a c a c a will be equal to multiplying c on each component c times a 1 c times a 2 c times a 3 so that if you do the cross product of ca plus b is going to be doing the determinant of i times j times k and then c times a1, c times a2, then c times a3 with b1, b2, b2 and g3...

Please give Ace some feedback