# Scalar Multiplication and Vector Addition

**scalar multiplication**and

**vector addition**. In general, when working with vectors numbers or constants are called

*.*

__scalars__Scalar Multiplication is when a vector is multiplied by a scalar (a number or a constant). If a vector

**v**is multiplied by a scalar k the result is k

**v**. If k is positive then k

**v**will have the same directions as

**v**. If k is negative, k

**v**will have the opposite direction as

**v**.

SCALAR MULTIPLICATION:

Let v = $\langle {v}_{1},{v}_{2}\rangle $ and k be a scalar.

k**v** = k$\langle {v}_{1},{v}_{2}\rangle $ = $\langle k{v}_{1},k{v}_{2}\rangle $

To add two vectors

**u**and

**v**, place the initial point of the second vector (without changing length or direction) on the terminal point of the first vector. Then join the initial point of the first vector to the terminus of the second vector. This joining line represents the sum of the two vectors.

The sum of vectors

**u**and

**v**in component form is:

ADDITION OF VECTORS:

Let u = $\langle {u}_{1},{u}_{2}\rangle $ and v = $\langle {v}_{1},{v}_{2}\rangle $

$u+v=\langle {u}_{1}+{v}_{1},{u}_{2}+{v}_{2}\rangle $

$u-v=u+\left(-v\right)=\langle {u}_{1}-{v}_{1},{u}_{2}-{v}_{2}\rangle $

Scalar multiplication and vector addition share the following properties:

PROPERTIES OF SCALAR MULTIPLICATION AND VECTOR ADDITION:*Let u,v and w be vectors and c and d be scalars.*

1.

**u**+

**v**=

**v**+

**u**2. (

**u**+

**v**) +

**w**=

**u**+ (

**v**+

**w**)

3.

**u**+ 0 =

**u**4.

**u**+ (-

**u**) = 0

5. c(d

**u**) = (cd)

**u**6. (c + d)

**u**= c

**u**+ d

**u**

7. c(

**u**+

**v**) = c

**u**+ c

**v**8. 1 ·

**u**=

**u**, 0 ·

**u**= 0

9. ||c

**v**|| = |c|||

**v**||

Let's look at a couple examples.

**Example 1: If $u=\langle -2,1\rangle $ and $v=\langle 7,-3\rangle $ find (a) u + v and (b) u - v.**

Add the x-component of both vectors. Do the same for the y-components. |
$u+v=\langle -2+7,1+\left(-3\right)\rangle $ $u+v=\langle 5,-2\rangle $ |

Remember u - v = u + (-v), therefore subtract the x-component of |
$u-v=u+\left(-v\right)=\langle -2-7,1-\left(-3\right)\rangle $ $u-v=\langle -9,4\rangle $ |

**Example 2: If $u=\langle 6,15\rangle $ and $v=\langle -5,20\rangle $ find (a) 2u + v and (b) 5u - 2v.**

a) First calculate 2
b) Next calculate 2 |
$2u=2\langle 6,15\rangle =\langle 2\xb76,2\xb715\rangle $ $2u=\langle 12,30\rangle $ $2u+v=\langle 12+\left(-5\right),30+20\rangle $ $2u+v=\langle 7,50\rangle $ |

a) First calculate 5
b) Next calculate 5 |
$5u=5\langle 6,15\rangle =\langle 5\xb76,5\xb715\rangle $ $5u=\langle 30,75\rangle $ $2v=2\langle -5,20\rangle =\langle 2\xb7\left(-5\right),2\xb720\rangle $ $2u=\langle -10,40\rangle $ $5u-2v=5u+\left(-2v\right)$ $5u-2v=\langle 30-\left(-10\right),75-40\rangle $ $5u-2v=\langle 40,35\rangle $ |

Related Links:Math algebra Unit Vectors Direction Angles of Vectors Pre Calculus |

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