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Concept: Multiplying a vector by a scalar quantity
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A scalar is a quantity which has the only magnitude $|m|$ in contrast to a vector which has both direction and magnitude $|m|angletheta$.
Intuitively, multiplying a vector by a scalar scales a vector by the respective magnitude. Moreover, various texts claim, that multiplying a vector by scalar only changes the magnitude but direction remains unaffected.
An unexpected result is noticed when we multiply a negative scalar by a vector. Notably, a vector $vec v = (x, y)$, when its multiplied by a negative scalar $-|q|$, it not only scales the magnitude by a quantity equals to $|q|$, but also flips the vector $-|q|vec v = (-|q|x, -|q|y) = |q(x^2+y^2)|angleleft(pi+arctanfracyxright)$.
Many notable texts, claim the behavior to be valid. But, I find a few contradictions
- If by multiplying a vector $vec v$ scales a vector, what is the intuition of negative scaling?
- If the magnitude is absolute, and scalar has the only magnitude, how can scalar be negative?
- If multiplying by a scalar does not change the direction, why is multiplying a vector by a negative quantity and thus flipping it, a valid behavior?
The way, I am trying to understand is, multiplying by a negative scalar is actually two operations.
- Multiplying the vector by (-1) which flips a vector. This generates the negative inverse of a vector $because vec v + (-1)times vec v = 0$
- Scale the resultant vector by the appropriate scalar quantity.
So, scalar -q is actually $|q| times -1$ and thus multiplying with the vector $vec v = |v|angletheta$
$$Rightarrow -q times vec v = |q| times -1 vec v = |q| times |v|angleleft(pi + thetaright)$$
But, I cannot find a source to substantiate my reasoning and need the community support to help me with my understanding.
linear-algebra vectors
asked Jul 26 at 19:52
Abhijit
2,337719
 |Â
show 1 more comment
up vote
0
down vote
favorite
A scalar is a quantity which has the only magnitude $|m|$ in contrast to a vector which has both direction and magnitude $|m|angletheta$.
Intuitively, multiplying a vector by a scalar scales a vector by the respective magnitude. Moreover, various texts claim, that multiplying a vector by scalar only changes the magnitude but direction remains unaffected.
An unexpected result is noticed when we multiply a negative scalar by a vector. Notably, a vector $vec v = (x, y)$, when its multiplied by a negative scalar $-|q|$, it not only scales the magnitude by a quantity equals to $|q|$, but also flips the vector $-|q|vec v = (-|q|x, -|q|y) = |q(x^2+y^2)|angleleft(pi+arctanfracyxright)$.
Many notable texts, claim the behavior to be valid. But, I find a few contradictions
- If by multiplying a vector $vec v$ scales a vector, what is the intuition of negative scaling?
- If the magnitude is absolute, and scalar has the only magnitude, how can scalar be negative?
- If multiplying by a scalar does not change the direction, why is multiplying a vector by a negative quantity and thus flipping it, a valid behavior?
The way, I am trying to understand is, multiplying by a negative scalar is actually two operations.
- Multiplying the vector by (-1) which flips a vector. This generates the negative inverse of a vector $because vec v + (-1)times vec v = 0$
- Scale the resultant vector by the appropriate scalar quantity.
So, scalar -q is actually $|q| times -1$ and thus multiplying with the vector $vec v = |v|angletheta$
$$Rightarrow -q times vec v = |q| times -1 vec v = |q| times |v|angleleft(pi + thetaright)$$
But, I cannot find a source to substantiate my reasoning and need the community support to help me with my understanding.
linear-algebra vectors
asked Jul 26 at 19:52
Abhijit
2,337719
I think when they say that multiplying by a scalar does not change the direction, what they mean is the span of the vector remains unchanged.
– gd1035
Jul 26 at 20:01
Multiplying by a negative does "flip" and scale the vector, but the span of the vector remains unchanged.
– gd1035
Jul 26 at 20:02
Multiplying by a negative number changes the magnitude and direction, as you say. I wouldn't worry too much about the words the text used to describe this.
– saulspatz
Jul 26 at 20:02
When they say a scalar is "a quantity that has magnitude but not direction", they are speaking roughly. The term "magnitude" in this context is not meant to imply that a scalar cannot be negative. A scalar can be negative, and that is fine. When they say that multiplying by a scalar "does not change direction", they are also speaking roughly, because multiplying by a negative scalar does reverse the direction.
– littleO
Jul 26 at 20:04
For a vector $v$: Magnitude: $|v|$, Direction: $O=rv: rtext scalar$, Orientation: Fixing a vector $win O$, with $wneq0$, $v$ is positively oriented with respect to $w$ if there is $r>0$ such that $rv=w$, negatively oriented with respect to $w$ if there is $r<0$ such that $rv=w$. With these definitions, multiplying $v$ by $-2$, changes magnitude, and orientation with respect to a fixed $w$, and not direction.
– user577471
Jul 26 at 20:11
 |Â
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
A scalar is a quantity which has the only magnitude $|m|$ in contrast to a vector which has both direction and magnitude $|m|angletheta$.
Intuitively, multiplying a vector by a scalar scales a vector by the respective magnitude. Moreover, various texts claim, that multiplying a vector by scalar only changes the magnitude but direction remains unaffected.
An unexpected result is noticed when we multiply a negative scalar by a vector. Notably, a vector $vec v = (x, y)$, when its multiplied by a negative scalar $-|q|$, it not only scales the magnitude by a quantity equals to $|q|$, but also flips the vector $-|q|vec v = (-|q|x, -|q|y) = |q(x^2+y^2)|angleleft(pi+arctanfracyxright)$.
Many notable texts, claim the behavior to be valid. But, I find a few contradictions
- If by multiplying a vector $vec v$ scales a vector, what is the intuition of negative scaling?
- If the magnitude is absolute, and scalar has the only magnitude, how can scalar be negative?
- If multiplying by a scalar does not change the direction, why is multiplying a vector by a negative quantity and thus flipping it, a valid behavior?
The way, I am trying to understand is, multiplying by a negative scalar is actually two operations.
- Multiplying the vector by (-1) which flips a vector. This generates the negative inverse of a vector $because vec v + (-1)times vec v = 0$
- Scale the resultant vector by the appropriate scalar quantity.
So, scalar -q is actually $|q| times -1$ and thus multiplying with the vector $vec v = |v|angletheta$
$$Rightarrow -q times vec v = |q| times -1 vec v = |q| times |v|angleleft(pi + thetaright)$$
But, I cannot find a source to substantiate my reasoning and need the community support to help me with my understanding.
linear-algebra vectors
asked Jul 26 at 19:52
Abhijit
2,337719
A scalar is a quantity which has the only magnitude $|m|$ in contrast to a vector which has both direction and magnitude $|m|angletheta$.
Intuitively, multiplying a vector by a scalar scales a vector by the respective magnitude. Moreover, various texts claim, that multiplying a vector by scalar only changes the magnitude but direction remains unaffected.
An unexpected result is noticed when we multiply a negative scalar by a vector. Notably, a vector $vec v = (x, y)$, when its multiplied by a negative scalar $-|q|$, it not only scales the magnitude by a quantity equals to $|q|$, but also flips the vector $-|q|vec v = (-|q|x, -|q|y) = |q(x^2+y^2)|angleleft(pi+arctanfracyxright)$.
Many notable texts, claim the behavior to be valid. But, I find a few contradictions
- If by multiplying a vector $vec v$ scales a vector, what is the intuition of negative scaling?
- If the magnitude is absolute, and scalar has the only magnitude, how can scalar be negative?
- If multiplying by a scalar does not change the direction, why is multiplying a vector by a negative quantity and thus flipping it, a valid behavior?
The way, I am trying to understand is, multiplying by a negative scalar is actually two operations.
- Multiplying the vector by (-1) which flips a vector. This generates the negative inverse of a vector $because vec v + (-1)times vec v = 0$
- Scale the resultant vector by the appropriate scalar quantity.
So, scalar -q is actually $|q| times -1$ and thus multiplying with the vector $vec v = |v|angletheta$
$$Rightarrow -q times vec v = |q| times -1 vec v = |q| times |v|angleleft(pi + thetaright)$$
But, I cannot find a source to substantiate my reasoning and need the community support to help me with my understanding.
linear-algebra vectors
asked Jul 26 at 19:52
Abhijit
2,337719
edited Jul 27 at 4:32
asked Jul 26 at 19:52
Abhijit
2,337719
asked Jul 26 at 19:52
Abhijit
2,337719
asked Jul 26 at 19:52
Abhijit
2,337719
2,337719
I think when they say that multiplying by a scalar does not change the direction, what they mean is the span of the vector remains unchanged.
– gd1035
Jul 26 at 20:01
Multiplying by a negative does "flip" and scale the vector, but the span of the vector remains unchanged.
– gd1035
Jul 26 at 20:02
Multiplying by a negative number changes the magnitude and direction, as you say. I wouldn't worry too much about the words the text used to describe this.
– saulspatz
Jul 26 at 20:02
When they say a scalar is "a quantity that has magnitude but not direction", they are speaking roughly. The term "magnitude" in this context is not meant to imply that a scalar cannot be negative. A scalar can be negative, and that is fine. When they say that multiplying by a scalar "does not change direction", they are also speaking roughly, because multiplying by a negative scalar does reverse the direction.
– littleO
Jul 26 at 20:04
For a vector $v$: Magnitude: $|v|$, Direction: $O=rv: rtext scalar$, Orientation: Fixing a vector $win O$, with $wneq0$, $v$ is positively oriented with respect to $w$ if there is $r>0$ such that $rv=w$, negatively oriented with respect to $w$ if there is $r<0$ such that $rv=w$. With these definitions, multiplying $v$ by $-2$, changes magnitude, and orientation with respect to a fixed $w$, and not direction.
– user577471
Jul 26 at 20:11
 |Â
show 1 more comment
I think when they say that multiplying by a scalar does not change the direction, what they mean is the span of the vector remains unchanged.
– gd1035
Jul 26 at 20:01
Multiplying by a negative does "flip" and scale the vector, but the span of the vector remains unchanged.
– gd1035
Jul 26 at 20:02
Multiplying by a negative number changes the magnitude and direction, as you say. I wouldn't worry too much about the words the text used to describe this.
– saulspatz
Jul 26 at 20:02
When they say a scalar is "a quantity that has magnitude but not direction", they are speaking roughly. The term "magnitude" in this context is not meant to imply that a scalar cannot be negative. A scalar can be negative, and that is fine. When they say that multiplying by a scalar "does not change direction", they are also speaking roughly, because multiplying by a negative scalar does reverse the direction.
– littleO
Jul 26 at 20:04
For a vector $v$: Magnitude: $|v|$, Direction: $O=rv: rtext scalar$, Orientation: Fixing a vector $win O$, with $wneq0$, $v$ is positively oriented with respect to $w$ if there is $r>0$ such that $rv=w$, negatively oriented with respect to $w$ if there is $r<0$ such that $rv=w$. With these definitions, multiplying $v$ by $-2$, changes magnitude, and orientation with respect to a fixed $w$, and not direction.
– user577471
Jul 26 at 20:11
I think when they say that multiplying by a scalar does not change the direction, what they mean is the span of the vector remains unchanged.
– gd1035
Jul 26 at 20:01
I think when they say that multiplying by a scalar does not change the direction, what they mean is the span of the vector remains unchanged.
– gd1035
Jul 26 at 20:01
Multiplying by a negative does "flip" and scale the vector, but the span of the vector remains unchanged.
– gd1035
Jul 26 at 20:02
Multiplying by a negative does "flip" and scale the vector, but the span of the vector remains unchanged.
– gd1035
Jul 26 at 20:02
Multiplying by a negative number changes the magnitude and direction, as you say. I wouldn't worry too much about the words the text used to describe this.
– saulspatz
Jul 26 at 20:02
Multiplying by a negative number changes the magnitude and direction, as you say. I wouldn't worry too much about the words the text used to describe this.
– saulspatz
Jul 26 at 20:02
When they say a scalar is "a quantity that has magnitude but not direction", they are speaking roughly. The term "magnitude" in this context is not meant to imply that a scalar cannot be negative. A scalar can be negative, and that is fine. When they say that multiplying by a scalar "does not change direction", they are also speaking roughly, because multiplying by a negative scalar does reverse the direction.
– littleO
Jul 26 at 20:04
When they say a scalar is "a quantity that has magnitude but not direction", they are speaking roughly. The term "magnitude" in this context is not meant to imply that a scalar cannot be negative. A scalar can be negative, and that is fine. When they say that multiplying by a scalar "does not change direction", they are also speaking roughly, because multiplying by a negative scalar does reverse the direction.
– littleO
Jul 26 at 20:04
For a vector $v$: Magnitude: $|v|$, Direction: $O=rv: rtext scalar$, Orientation: Fixing a vector $win O$, with $wneq0$, $v$ is positively oriented with respect to $w$ if there is $r>0$ such that $rv=w$, negatively oriented with respect to $w$ if there is $r<0$ such that $rv=w$. With these definitions, multiplying $v$ by $-2$, changes magnitude, and orientation with respect to a fixed $w$, and not direction.
– user577471
Jul 26 at 20:11
For a vector $v$: Magnitude: $|v|$, Direction: $O=rv: rtext scalar$, Orientation: Fixing a vector $win O$, with $wneq0$, $v$ is positively oriented with respect to $w$ if there is $r>0$ such that $rv=w$, negatively oriented with respect to $w$ if there is $r<0$ such that $rv=w$. With these definitions, multiplying $v$ by $-2$, changes magnitude, and orientation with respect to a fixed $w$, and not direction.
– user577471
Jul 26 at 20:11
 |Â
show 1 more comment
2 Answers
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oldest
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up vote
1
down vote
We have that for $lambdain mathbbR$
for $lambda>0$ we have that the operation $lambdavec v$ scale $vec v$ of a factor $|lambda|$ with the same orientation
for $lambda<0$ we have that the operation $lambdavec v$ scale $vec v$ of a factor $|lambda|$ with the opposite orientation
Note that in both cases the direction doesn't change since both vectors belong to the same line.
answered Jul 26 at 20:11
gimusi
65k73583
add a comment |Â
up vote
1
down vote
In a real vector space where the scalars are real numbers, a scalar has a magnitude and a sign.
Thus when you multiply a vector by a scalar you scale the magnitude of the vector and change or not change the direction of your vector based on the sign of the scalar.
( The vector stays on the same line, which sometimes is interpreted as not changing the direction )
The concept is clear but the terminology is sometimes confusing.
answered Jul 26 at 20:12
Mohammad Riazi-Kermani
27.3k41851
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
We have that for $lambdain mathbbR$
for $lambda>0$ we have that the operation $lambdavec v$ scale $vec v$ of a factor $|lambda|$ with the same orientation
for $lambda<0$ we have that the operation $lambdavec v$ scale $vec v$ of a factor $|lambda|$ with the opposite orientation
Note that in both cases the direction doesn't change since both vectors belong to the same line.
answered Jul 26 at 20:11
gimusi
65k73583
add a comment |Â
up vote
1
down vote
We have that for $lambdain mathbbR$
for $lambda>0$ we have that the operation $lambdavec v$ scale $vec v$ of a factor $|lambda|$ with the same orientation
for $lambda<0$ we have that the operation $lambdavec v$ scale $vec v$ of a factor $|lambda|$ with the opposite orientation
Note that in both cases the direction doesn't change since both vectors belong to the same line.
answered Jul 26 at 20:11
gimusi
65k73583
add a comment |Â
up vote
1
down vote
up vote
1
down vote
We have that for $lambdain mathbbR$
for $lambda>0$ we have that the operation $lambdavec v$ scale $vec v$ of a factor $|lambda|$ with the same orientation
for $lambda<0$ we have that the operation $lambdavec v$ scale $vec v$ of a factor $|lambda|$ with the opposite orientation
Note that in both cases the direction doesn't change since both vectors belong to the same line.
answered Jul 26 at 20:11
gimusi
65k73583
We have that for $lambdain mathbbR$
for $lambda>0$ we have that the operation $lambdavec v$ scale $vec v$ of a factor $|lambda|$ with the same orientation
for $lambda<0$ we have that the operation $lambdavec v$ scale $vec v$ of a factor $|lambda|$ with the opposite orientation
Note that in both cases the direction doesn't change since both vectors belong to the same line.
answered Jul 26 at 20:11
gimusi
65k73583
answered Jul 26 at 20:11
gimusi
65k73583
answered Jul 26 at 20:11
gimusi
65k73583
answered Jul 26 at 20:11
gimusi
65k73583
65k73583
add a comment |Â
add a comment |Â
up vote
1
down vote
In a real vector space where the scalars are real numbers, a scalar has a magnitude and a sign.
Thus when you multiply a vector by a scalar you scale the magnitude of the vector and change or not change the direction of your vector based on the sign of the scalar.
( The vector stays on the same line, which sometimes is interpreted as not changing the direction )
The concept is clear but the terminology is sometimes confusing.
answered Jul 26 at 20:12
Mohammad Riazi-Kermani
27.3k41851
add a comment |Â
up vote
1
down vote
In a real vector space where the scalars are real numbers, a scalar has a magnitude and a sign.
Thus when you multiply a vector by a scalar you scale the magnitude of the vector and change or not change the direction of your vector based on the sign of the scalar.
( The vector stays on the same line, which sometimes is interpreted as not changing the direction )
The concept is clear but the terminology is sometimes confusing.
answered Jul 26 at 20:12
Mohammad Riazi-Kermani
27.3k41851
add a comment |Â
up vote
1
down vote
up vote
1
down vote
In a real vector space where the scalars are real numbers, a scalar has a magnitude and a sign.
Thus when you multiply a vector by a scalar you scale the magnitude of the vector and change or not change the direction of your vector based on the sign of the scalar.
( The vector stays on the same line, which sometimes is interpreted as not changing the direction )
The concept is clear but the terminology is sometimes confusing.
answered Jul 26 at 20:12
Mohammad Riazi-Kermani
27.3k41851
In a real vector space where the scalars are real numbers, a scalar has a magnitude and a sign.
Thus when you multiply a vector by a scalar you scale the magnitude of the vector and change or not change the direction of your vector based on the sign of the scalar.
( The vector stays on the same line, which sometimes is interpreted as not changing the direction )
The concept is clear but the terminology is sometimes confusing.
answered Jul 26 at 20:12
Mohammad Riazi-Kermani
27.3k41851
answered Jul 26 at 20:12
Mohammad Riazi-Kermani
27.3k41851
answered Jul 26 at 20:12
Mohammad Riazi-Kermani
27.3k41851
answered Jul 26 at 20:12
Mohammad Riazi-Kermani
27.3k41851
27.3k41851
add a comment |Â
add a comment |Â
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I think when they say that multiplying by a scalar does not change the direction, what they mean is the span of the vector remains unchanged.
– gd1035
Jul 26 at 20:01
Multiplying by a negative does "flip" and scale the vector, but the span of the vector remains unchanged.
– gd1035
Jul 26 at 20:02
Multiplying by a negative number changes the magnitude and direction, as you say. I wouldn't worry too much about the words the text used to describe this.
– saulspatz
Jul 26 at 20:02
When they say a scalar is "a quantity that has magnitude but not direction", they are speaking roughly. The term "magnitude" in this context is not meant to imply that a scalar cannot be negative. A scalar can be negative, and that is fine. When they say that multiplying by a scalar "does not change direction", they are also speaking roughly, because multiplying by a negative scalar does reverse the direction.
– littleO
Jul 26 at 20:04
For a vector $v$: Magnitude: $|v|$, Direction: $O=rv: rtext scalar$, Orientation: Fixing a vector $win O$, with $wneq0$, $v$ is positively oriented with respect to $w$ if there is $r>0$ such that $rv=w$, negatively oriented with respect to $w$ if there is $r<0$ such that $rv=w$. With these definitions, multiplying $v$ by $-2$, changes magnitude, and orientation with respect to a fixed $w$, and not direction.
– user577471
Jul 26 at 20:11