Vector-Vector Product - Linear Algebra
Card 0 of 184
Compute
, where


Compute , where
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Before we compute the product of
, and
, we need to check if it is possible to take the product. We will check the dimensions.
is
, and
is
, so the dimensions of the resulting matrix will be
. Now let's compute it.

Before we compute the product of , and
, we need to check if it is possible to take the product. We will check the dimensions.
is
, and
is
, so the dimensions of the resulting matrix will be
. Now let's compute it.
Find the vector-vector product of the following vectors.
![A=[1 \ 2]](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/727325/gif.latex)

Find the vector-vector product of the following vectors.
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Calculate
, given


Calculate , given
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By definition,
.
By definition,
.
What is the physical significance of the resultant vector
, if
?
What is the physical significance of the resultant vector , if
?
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By definition, the resultant cross product vector (in this case,
) is orthogonal to the original vectors that were crossed (in this case,
and
). In
, this means that
is a vector that is normal to the plane containing
and
.
By definition, the resultant cross product vector (in this case, ) is orthogonal to the original vectors that were crossed (in this case,
and
). In
, this means that
is a vector that is normal to the plane containing
and
.
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Let
and
be vectors defined by
.
Find the dot product
.
Let and
be vectors defined by
.
Find the dot product .
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Vectors
and
are both of length 4. The dimensions match and the dot product exists.
![\bf{a}\cdot\bf{b} = [ 9, 0, 9, 5]\cdot [4, 3, 0, 9] = (9)(4)+(0)(3)+(9)(0)+(5)(9)=81](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/939943/gif.latex)
Vectors and
are both of length 4. The dimensions match and the dot product exists.
Let
and
be vectors defined by
.
Find the cross product
.
Let and
be vectors defined by
.
Find the cross product .
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We can find the cross product by calculating the determinant of the following matrix


We can find the cross product by calculating the determinant of the following matrix
Let
and
be vectors defined by
.
Find the cross product
.
Let and
be vectors defined by
.
Find the cross product .
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We find the cross product by finding the determinant of the following matrix


We find the cross product by finding the determinant of the following matrix


The expression
yields a polynomial of what degree?
The expression yields a polynomial of what degree?
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The dot product
is the sum of the products of entries in corresponding positions, so


The degree of a term of a polynomial is the sum of the exponents of its variables. Each term in this polynomial has exponent sum 5, so each term has degree 5. The degree of the polynomial is the greatest of the degrees, so the polynomial has degree 5.
The dot product is the sum of the products of entries in corresponding positions, so
The degree of a term of a polynomial is the sum of the exponents of its variables. Each term in this polynomial has exponent sum 5, so each term has degree 5. The degree of the polynomial is the greatest of the degrees, so the polynomial has degree 5.