If \vec{w} = \vec{u} \times \vec{v}
, explain why \vec{w} \cdot \vec{u}, \vec{w} \cdot \vec{v}
, and \vec{w} \cdot (a\vec{u} + b\vec{v})
are all zero.
State whether the following expressions are vectors, scalars, or meaningless.
a. \vec{a} \cdot (\vec{b} \times \vec{c})
b. (\vec{a} \cdot \vec{b}) \times (\vec{b} \cdot \vec{c})
State whether the following expressions are vectors, scalars, or meaningless.
(\vec{a} \times \vec{b})\cdot (\vec{b} \times \vec{c})
State whether the following expressions are vectors, scalars, or meaningless.
(\vec{a} \times \vec{b}) + (\vec{b} \times \vec{c})
State whether the following expressions are vectors, scalars, or meaningless.
(\vec{a} \cdot \vec{b}) + (\vec{b} \cdot \vec{c})
State whether the following expressions are vectors, scalars, or meaningless.
(\vec{a} \cdot \vec{b}) -\vec{c}
Use the cross product to find a vector perpendicular to each of the following pairs of vectors. Check your answer using the dot product.
(4, 0, 0)
and (0, 0, 4)
Use the cross product to find a vector perpendicular to each of the following pairs of vectors. Check your answer using the dot product.
(1, 2, 1)
and (6, 0, 6)
Use the cross product to find a vector perpendicular to each of the following pairs of vectors. Check your answer using the dot product.
(2, -1, 3)
and (1, 4, -2)
Use the cross product to find a vector perpendicular to each of the following pairs of vectors. Check your answer using the dot product.
(0, 2, -5)
and (-4, 9, 0)
Find the unit vector perpendicular to \vec{a} = (4, -3, 1)
and \vec{b}= (2, 3, -1)
.
Find two vectors perpendicular to both (3, -6, 3)
and (-2, 4, 2)
.
Express the unit vectors \vec{i}, \vec{j}
, and \vec{k}
as ordered triples and show that
a. \vec{i} \times \vec{j} = \vec{k}
b. \vec{k} \times \vec{j} = -\vec{i}
Using components, show that
\vec{u} \times \vec{v} = -\vec{v} \times \vec{u}
for any vectors \vec{u}
and \vec{v}
Using components, show that
\vec{u} \times \vec{v} = \vec{0}
if \vec{u}
and \vec{v}
are collinear
Prove that \displaystyle
|\vec{a} \times \vec{b}| = \sqrt{(\vec{a} \cdot \vec{a})(\vec{b}\cdot \vec{b}) - (\vec{a}\cdot \vec{b})^2}
.
Given \vec{a} = (2, 1, 0), \vec{b} = (-1, 0, 3)
, and \vec{c} = (4, -1, 1)
, calculate the following triple scalar and triple vector products.
\vec{a} \times \vec{b} \cdot \vec{c}
Given \vec{a} = (2, 1, 0), \vec{b} = (-1, 0, 3)
, and \vec{c} = (4, -1, 1)
, calculate the following triple scalar and triple vector products.
\vec{b} \times \vec{c} \cdot \vec{a}
Given \vec{a} = (2, 1, 0), \vec{b} = (-1, 0, 3)
, and \vec{c} = (4, -1, 1)
, calculate the following triple scalar and triple vector products.
\vec{a} \times \vec{b} \times \vec{c}
Given two non-collinear vectors \vec{a}
and \vec{b}
, show that \vec{a}, \vec{a} \times \vec{b}
, and (\vec{a} \times \vec{b})\times \vec{a}
are mutually perpendicular.
Prove that the triple scalar product of the vectors \vec{u}, \vec{v}
, and \vec{w}
has the property that \vec{u} \cdot (\vec{v} \times \vec{w}) = (\vec{u} \times \vec{v}) \cdot \vec{w}
. Carry out the proof by expressing both sides of the equation in terms of components of the vectors.
If the cross product of \vec{a}
and \vec{b}
is equal to the cross product of \vec{a}
and \vec{c}
, this does not necessarily mean that \vec{b}
equals \vec{c}
. Show why this is so
a. by making an algebraic argument.
b. by drawing a geometrical diagram.
a. If \vec{a} = (1, 3, -1), \vec{b} = (2, 1, 5), \vec{v} = (-3, y, z)
, and \vec{a} \times \vec{v} = \vec{b}
, find y
and z
.
a. Find another vector \vec{v}
fo which \vec{a} \times \vec{v} = \vec{b}
.
c. Explain why there are infinitely many vectors \vec{v}
fo which \vec{a} \times \vec{v} = \vec{b}
.
Introductio to Cross Product
Proof of Algebraic Formula for Cross Product
Finding Cross Product between Geometric Vectors
Finding Cross product between Cartesian Vectors
Finding Area of Triangle in Cartesian Coordinates
Coplanar of Vectors