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목차

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Equal(=)

This is a simple and basic property ****of vectors.

• If two vector’s components are equal(=), two vectors are considered equal(=).
  • $\vec{u}=(u_x,u_y,u_z)$
  • $\vec{v}=(v_x,v_y,v_z)$
  • If $\vec{u}=(1,2,3)$ and $\vec{v}=(1,2,3)$, then $\vec{u}=\vec{v}$

Add(+)

This is a simple operation to add the components of two vectors.

$$ \vec{u}+\vec{v}=(u_x+v_x,u_y+v_y,u_z+v_z) $$

• $\vec{u}=(1,2,3)$
• $\vec{v}=(4,5,6)$
• $\vec{u}+\vec{v}=(5,7,9)$

Geometric Interpretation

To add two vectors geometrically.

• Place tail of $\vec{A}$ at head of $\vec{B}$.
• $\vec{A} + \vec{B}$ : starts at tail of $\vec{A}$ &ends at head $\vec{B}$.

Physically

• Adding two forces (vectors) in the same direction results in a stronger force (a longer vector).
• Adding two forces in opposite directions results in a weaker force (a shorter vector).

Subtract(-)

This is a simple operation to subtract the components of two vectors.

$$ \vec{u}-\vec{v}=(u_x-v_x,u_y-v_y,u_z-v_z) $$

• $\vec{u}=(1,2,3)$
• $\vec{v}=(4,5,6)$
• $\vec{u}-\vec{v}=(-3,-3,-3)$

Geometric Interpretation

To subtract two vectors geometrically. subtraction operation is very similar to addition.

• Place tail of $\vec{A}$ at tail of $\vec{B}$.
• $\vec{A} - \vec{B}$ : starts at head of $\vec{B}$ & ends at head of $\vec{A}$.

Physically

• If you think of vectors as points in space
  • $\vec{v}-\vec{u}$ represents the direction & distance from point $\vec{u}$ to point $\vec{v}$.

Scalar Mulitiply(*)

Scalar multiplication involves multiplying each component of a vector by a scalar.

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Scalar

• A scalar is a single number (real number). </aside>

Fomular

$$ c\vec{v}=(cv_x,cv_y,cv_z) $$

• $\vec{v}=(2,3,4)$
• If $c=2$, then $2\vec{v}=(4,6,8)$
• If $c=-1$, then $-\vec{v}=(-2,-3,-4)$

Geometric Interpretation

• Scalar multiplication by $c$ changes the length (magnitude) of a vector:
  • If $c>0$, the direction is same, length is multiplied
  • If $c<0$, the direction is flipped, length is multiplied