But, as I said, you don't actually need the coordinates at all.

In summary, using the transformation ##u \times v = -\vec v \cdot [\vec u (\vec u \cdot \vec v)- \vec v (\vec u^2)]##, the equation for the vector sum of three vectors can be written in a more general form as:$$(\vec u + \vec v + \vec w)^2 = 2+3+6
  • #1
hnnhcmmngs
19
0

Homework Statement



Calculate |u+v+w|, knowing that u, v, and w are vectors in space such that
|u|=√2, |v|=√3, u is perpendicular to v, w=u×v.

Homework Equations



|w|=|u×v|=|u|*|v|*sinΘ

The Attempt at a Solution


[/B]
Θ=90°
|w|=(√2)*(√3)*sin(90°)=√(6)

Then I tried to use
u={√2,0,0}
v={0,√3,0}
w={0,0,√6}
and I got that |u+v+w|=√(2+3+6)=√11
but I'm trying to find a way to do this problem where I don't assume that those are the vectors.
 
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  • #2
hnnhcmmngs said:

Homework Statement



Calculate |u+v+w|, knowing that u, v, and w are vectors in space such that
|u|=√2, |v|=√3, u is perpendicular to v, w=u×v.

Homework Equations



|w|=|u×v|=|u|*|v|*sinΘ

The Attempt at a Solution


[/B]
Θ=90°
|w|=(√2)*(√3)*sin(90°)=√(6)

Then I tried to use
u={√2,0,0}
v={0,√3,0}
w={0,0,√6}
and I got that |u+v+w|=√(2+3+6)=√11
but I'm trying to find a way to do this problem where I don't assume that those are the vectors.

There is no loss of generality in what you did. Assuming that ##u,v,w## are vectors in some Cartesian coordinate system ##(x,y,z)##, just change coordinates to a new system ##(x', y', z')## in which ##u## points along the ##x'##-axis, ##v## points along the ##y'##-axis and ##w## points along the ##z'##-axis. That takes you right back to your original calculations.

Note added in edit: that argument assumes that the cross-product ##u \times v## behaves like a vector under a rotation; that is, if ##{\cal R}## is a rotation, then ##{\cal R}(u \times v) = {\cal R} u \: \times \: {\cal R} v.## (Here I mean that ##{\cal R} u## is the vector ##u## re-expressed in a new coordinate system obtained by applying ##{\cal R}## to the original unit coordinate vectors ##e_x, e_y, e_z##; it does not mean that the "physical" vector ##u## is rotated.) That "rotation" result is true, but needs to be proved, a task I will leave to you.
 
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  • #3
However, it should be noted that you can do it without ever referring to a coordinate system at all, just square and expand the vector sum:
$$
(\vec u + \vec v + \vec w)^2 = \vec u^2 + \vec v^2 + \vec w^2 + 2 (\vec u \cdot \vec v + \vec u \cdot \vec w + \vec v \cdot \vec w).
$$
Now, all of the inner products in the parenthesis are zero because all of the vectors are orthogonal. Furthermore
$$
\vec w^2 = (\vec u \times \vec v)^2 = (\vec u \times \vec v)\cdot(\vec u \times \vec v)
= -\vec v \cdot [\vec u \times (\vec u \times \vec v )].
$$
Apply the BAC-CAB rule:
$$
\vec w^2 = -\vec v \cdot [\vec u (\vec u \cdot \vec v)- \vec v (\vec u^2)] = \vec v^2 \vec u^2 = 6.
$$
It follows that
$$
(\vec u + \vec v + \vec w)^2 = 2+3+6 = 11.
$$
 

Related to But, as I said, you don't actually need the coordinates at all.

1. What is a vector?

A vector is a mathematical quantity that has both magnitude (size) and direction. It is often represented graphically as an arrow, with the length of the arrow representing the magnitude and the direction of the arrow indicating the direction.

2. How do you add two or more vectors?

To add two or more vectors, you must first determine their components (magnitude and direction) and then use vector addition rules to find the resultant vector. This can be done graphically or algebraically, depending on the given information.

3. What is the difference between vector addition and scalar addition?

Vector addition involves adding two or more vectors together, taking into account their magnitude and direction. Scalar addition, on the other hand, involves adding two or more scalar quantities (numbers) together, without considering direction.

4. Can vectors be multiplied?

Yes, vectors can be multiplied, but not in the traditional sense of multiplication. There are two types of vector multiplication: dot product and cross product. The dot product results in a scalar quantity, while the cross product results in a vector quantity.

5. How are vectors used in real life?

Vectors are used in various fields, including physics, engineering, and navigation. They are used to represent and analyze forces, velocities, and displacements. For example, vectors are used in calculating the trajectory of a projectile, determining the direction and speed of a moving object, and designing structures that can withstand different forces.

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