- #1
werson tan
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Is the Angle Between Vector A and Ax Less Than 90 Degrees When Alpha Exceeds 90?
In summary: A_{yz}## is a vector with a magnitude and a direction that is the same as ##\vec A## but with the y-component replaced by ##\vec A_y## (so ##\vec A_{yz} = \vec A_y \, \hat\imath\ ##).In summary, the angle between the force and the positive x-axis is more than 90 degree, meaning that the resultant angle between the force and the line from the origin to A is less than 90 degree.
![DSC_0092[1].JPG](/data/attachments/73/73144-e38285b6979df3f047bcd05aa39df95f.jpg)
![DSC_0093[1].JPG](/data/attachments/73/73145-daac108dc9b1e7f817ff409380a87aff.jpg)
- #2
BvU
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No. It will be exactly 90##^\circ##
(the 90##^\circ## as indicated in figure 2-26 (a) )
(the 90##^\circ## as indicated in figure 2-26 (a) )
- #3
werson tan
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why , i can't visualize it??BvU said:
- #4
werson tan
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If the force is lean towards negative x-axis , it means the angle between the force and positive x -axis is more than 90 degree, right ? then how is it possible for it to remain 90 degree?BvU said:
- #5
BvU
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You were talking about the line from Ax to A, not about the line from the origin to A (i.e. ##\vec A## itself).
- #6
werson tan
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ya , i mean the line from Ax to A, not about the line from the origin to A (i.e. A⃗ itself).BvU said:
if the force is lean towards the negative x-axis then the , we would have three line , right , namely the line joining origin to Ax , A(which is lean towards negative x-axis ) , and a line which join A to negative x-axis , right ? if it is so , then the i an undersatnd why the angle is 90 degree. if it so , the triangle formed is at opposite of the current one .
- #7
BvU
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I think you've got it all right and correct, just that the wording in post #1 was rather unfortunate.
With nowadays visualization possibilities it's almost a stone-age approach to put things in words, but let's do it anyway:
let's look in the blue plane of figure 2-26 (a) for the case ##\alpha \in [\pi/2, \pi]## where all the angles in our story are
so the vertical line Ayz is in the yz plane and that plane is perpendicular to the x axis.
and as you can see the line from Ax to A (the green line) is parallel to that plane and therefore also perpendicular to the x axis. Hence my 90 ##^\circ##.
You of course in your post #1 meant to refer to the angle between ##\vec A## and ##\vec A_x## which as you can see is ##\pi-\alpha ## and thereby in the range 0 to 90##^\circ##.
With nowadays visualization possibilities it's almost a stone-age approach to put things in words, but let's do it anyway:
let's look in the blue plane of figure 2-26 (a) for the case ##\alpha \in [\pi/2, \pi]## where all the angles in our story are
[edit] I mean to say that all the angles are in that plane. Sorry for the ambiguity.
so the vertical line Ayz is in the yz plane and that plane is perpendicular to the x axis.
and as you can see the line from Ax to A (the green line) is parallel to that plane and therefore also perpendicular to the x axis. Hence my 90 ##^\circ##.
You of course in your post #1 meant to refer to the angle between ##\vec A## and ##\vec A_x## which as you can see is ##\pi-\alpha ## and thereby in the range 0 to 90##^\circ##.
Note that I forgot to draw the arrow above ##\vec A_x## and ##\vec A_{yz}##. Nobody is perfect, be we keep trying.
To be specific: in ##\vec A_x = A_x\, \hat\imath\ ##, ##\ A_x## is a number (negative for the alpha in the picture: ##A_x = |\vec A | \cos\alpha##
To be specific: in ##\vec A_x = A_x\, \hat\imath\ ##, ##\ A_x## is a number (negative for the alpha in the picture: ##A_x = |\vec A | \cos\alpha##
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FAQ: Is the Angle Between Vector A and Ax Less Than 90 Degrees When Alpha Exceeds 90?
What is the definition of "direction of cartesian vector"?
The direction of a cartesian vector refers to the orientation or angle at which the vector is pointing in relation to the coordinate axes. It is often represented by an angle measured from the positive x-axis in a counterclockwise direction.
How is the direction of a cartesian vector represented?
The direction of a cartesian vector is typically represented by an angle, either in degrees or radians, measured from the positive x-axis. It can also be represented by a unit vector, which is a vector with a magnitude of 1 in the same direction as the original vector.
Can the direction of a cartesian vector be negative?
No, the direction of a cartesian vector is always positive and is measured in a counterclockwise direction from the positive x-axis. However, if the vector is pointing in the opposite direction, it can be represented by a negative angle.
How is the direction of a cartesian vector calculated?
The direction of a cartesian vector can be calculated by using trigonometric functions such as sine, cosine, and tangent. The angle can be determined by taking the inverse of these functions using the x and y components of the vector.
What is the significance of the direction of a cartesian vector?
The direction of a cartesian vector is important in determining the overall movement or force of an object. It also helps in understanding the relationship between different vectors and their effects on a system. Additionally, it is essential in solving problems involving vector addition and manipulation.