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nick227
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my teacher told me to think about this and i don't seem to get it. given vectors X,Y,& Z; is there geometric significance when X*(YxZ)=0
* is dot product and x is cross product
* is dot product and x is cross product
HallsofIvy said:The length of Y x Z is the area of a parallelogram with adjacent sides given by Y and Z.
X*(YxZ) is the the volume of a parallopiped having sides, at one vertex, given by X, Y, and Z. You can see that by using the formulas [itex]X*Y= |X||Y|sin(\theta)[/itex] and length of [itex]X x Y= |X||Y|cos(\theta)[/itex].
HallsofIvy said:The length of Y x Z is the area of a parallelogram with adjacent sides given by Y and Z.
X*(YxZ) is the the volume of a parallopiped having sides, at one vertex, given by X, Y, and Z. You can see that by using the formulas [itex]X*Y= |X||Y|sin(\theta)[/itex] and length of [itex]X x Y= |X||Y|cos(\theta)[/itex].
nick227 said:isn't [itex]X x Y= |X||Y|sin(\theta)[/itex] and [itex]X*Y= |X||Y|cos(\theta)[/itex]?
robphy said:You are (mostly) correct.
It is [itex] | \vec X \times \vec Y | = |\vec X| |\vec Y| |\sin\theta| [/itex] and [itex] \vec X \cdot \vec Y = |\vec X| |\vec Y| \cos\theta [/itex], where [itex]\theta[/tex] is the angle between the vectors. ([itex] \vec X \times \vec Y [/itex] is a vector with magnitude [itex] |\vec X| |\vec Y| |\sin\theta| [/itex] with direction perpendicular to the plane determined by [itex]\vec X [/itex] and [itex]\vec Y [/itex], according to the right-hand-rule.)
(I suspect HallsofIvy's typo was due to a confusion over the symbols " * " and its synonym " X " for multiplication.)
HallsofIvy gave the interpretation of the (scalar-)triple-product as the volume of a parallelopiped (a generally-slanted box with parallel sides) formed with those vectors. How would you describe this box if its volume were zero? What does that tell you about the relationship between [itex]\vec X[/itex], [itex]\vec Y[/itex] and [itex]\vec Z[/itex], along the lines of cornfall's suggestion?
nick227 said:if the volume of the box is zero, then its not a 3d figure, its 2d. in that case, its a square.
robphy said:So, what does that mean for vectors X, Y, and Z?
nick227 said:is it that all three vectors are on the same plane?
robphy said:ah... you changed your answer on me.
That's correct.
HallsofIvy said:The length of Y x Z is the area of a parallelogram with adjacent sides given by Y and Z.
X*(YxZ) is the the volume of a parallopiped having sides, at one vertex, given by X, Y, and Z. You can see that by using the formulas [itex]X*Y= |X||Y|sin(\theta)[/itex] and length of [itex]X x Y= |X||Y|cos(\theta)[/itex].
Yes to the last- that's exactly what I said. No to the first. X x Y is a vector, not a number and what you give is its length.nick227 said:isn't [itex]X x Y= |X||Y|sin(\theta)[/itex] and [itex]X*Y= |X||Y|cos(\theta)[/itex]?
What you wrote makes no sense- you cannot take the the cross product of two numbers!also, from that definition, would the angle between Y and Z equal to 0? when i break the given down, i get:
|X||Y|cos(theta) x |X||Z|cos(theta).
i can't really see the geometric significance.
A vector is a mathematical quantity that has both magnitude (size) and direction. It is commonly represented by an arrow pointing in the direction of the vector and labeled with its magnitude.
A scalar only has magnitude, while a vector has both magnitude and direction. For example, temperature is a scalar quantity as it only has a value (magnitude), while velocity is a vector quantity as it has both a value (magnitude) and direction.
Vectors can be added or subtracted using the head-to-tail method, where the tail of one vector is placed at the head of another vector. The resulting vector is the one that connects the tail of the first vector to the head of the last vector. The magnitude and direction of the resulting vector can be determined using trigonometric functions.
The dot product of two vectors is a mathematical operation that results in a scalar. It is calculated by multiplying the magnitudes of the two vectors and the cosine of the angle between them. It can be used to find the angle between two vectors or to project one vector onto another.
Vectors are commonly used in mathematics, physics, engineering, and computer science. They are also used in many real-world applications, such as navigation, graphics, and data analysis.