All Linear Algebra Resources
Example Questions
Example Question #33 : Vector Vector Product
Which of the following applies to , where " " and "" refer to the dot product and the cross product of two vectors?
is an undefined expression.
The cross product of two vectors in is also a vector in . It follows that and ; it further follows that.
Example Question #34 : Vector Vector Product
;
Which of the following applies to , where "" refers to the cross product of two vectors, and "" refers to either scalar or vector addition, as applicable?
is an undefined expression.
is an undefined expression.
The cross product of two vectors is defined only if both vectors are in . and are vectors in , so is undefined; consequently, so is .
Example Question #35 : Vector Vector Product
;
Which of the following applies to , where " " refers to the dot product of two vectors, and "" refers to either scalar or vector addition, as applicable?
is an undefined expression.
The dot product of two vectors in the same vector space is a scalar quantity. , so and are in the same vector space; their dot product is defined, and . For similar reasons, . Therefore, their sum is defined, and .
Example Question #35 : Vector Vector Product
.
Which of the following applies to , where " " and "" refer to the dot product and the cross product of two vectors, and "" refers to either scalar or vector addition, as applicable?
is an undefined expression.
is an undefined expression.
The cross product of two vectors in is also a vector in . The dot product of two such vectors is a scalar. Since a vector and a scalar cannot be added, is an undefined expression.
Example Question #161 : Matrices
True or false: It follows that .
True
False
True
One property of vector dot products is commutativity - that is,
.
Therefore, if , then .
The statement is true,
Example Question #162 : Matrices
True or false: It follows that .
False
True
False
One property of vector cross products is anticommutativity - that is,
.
If , it follows that
.
The statement is false.
Example Question #38 : Vector Vector Product
Which of the following applies to , where " " and "" refer to the dot product and the cross product of two vectors?
is an undefined expression.
The cross product of two vectors in is also a vector in . It follows that and . The dot product of two vectors in the same vector space, is a scalar, so , the dot product of two vectors in , is a scalar in .
Example Question #40 : Vector Vector Product
Find and so that .
There does not exist any such and
The cross product of two vectors in can be set up and calculated as if it were a determinant of a matrix with its top row comprising , the unit vectors of , and the other two comprising the elements of and :
Calculate as you would a determinant, adding the upper-left to lower-right products and subtracting upper-right to lower-left products:
The cross-product is equal to
We want this vector to be equal to , so the following must hold:
Examining the third equation, , we find that this is consistent with the other equations, since and make this true. Therefore, and are the values sought.
Example Question #41 : Vector Vector Product
.
Which of the following holds for ?
Note: (boldface) refers to the vector ;
, but the result need not be .
, but the result need not be 0.
is an undefined expression.
, where 0 refers to the real zero.
, where refers to the vector .
, where refers to the vector .
One property of vector cross products is anticommutativity - that is,
.
It therefore follows that for all ,
, the zero vector .
Example Question #42 : Vector Vector Product
A parallelogram has these two vectors as sides. Find so that the parallelogram is a rhombus.
For the parallelogram formed by and to be a rhombus, the vectors must be of equal length, or norm - . The norm of a vector is equal to the square root of the sum of the squares of its entries; to set the norms equal, it suffices to set the squares of the norms equal, and to solve for :
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