# The sum of A and B is 5i+j and their difference is 3i-j what is vector A?

Jun 28, 2018

$i$

#### Explanation:

vector $A$ can be written as $a i + b j$ where $a$ and $b$ are two unknowns.

vector $B$ can be written as $c i + \mathrm{dj}$ where $c$ and $d$ are two unknowns.

$\left(a i + b j\right) + \left(c i + \mathrm{dj}\right) = \left(a + c\right) i + \left(b + d\right) j = 5 i + j$

$a + c = 5 , b + d = 1$

$\left(a i + b j\right) - \left(c i + \mathrm{dj}\right) = \left(a - c\right) i + \left(b - d\right) j = 3 i - j$

$a - c = 3 , b - d = - 1$

$a + c = 5 , a - c = 3$

$a + c = \left(a - c\right) + 8$
$a + c = a - c + 8$
$c = - c + 8$
$2 c = 8$
$c = 4$

$a + c = 5$
$a + 4 = 5$
$a = 1$

$b + d = 1 , b - d = - 1$

$b + d = \left(b - d\right) + 2$
$b + d = b - d + 2$
$d = - d + 2$
$2 d = 2$
$d = 1$

$b + d = 1$
$b + 1 = 1$
$b = 0$

hence, we have
$a = 1$
$b = 0$

$c = 4$
$d = 1$

vector $A = a i + b j$
substituting the $a$ and $b$ values in gives $i + 0$, or $i$.

Jun 28, 2018

The sum of A and B is 5i+j

• $\boldsymbol{A} + \boldsymbol{B} = 5 \boldsymbol{i} + \boldsymbol{j} q \quad \square$

"and their difference is 3i-j"

• $\boldsymbol{A} - \boldsymbol{B} = 3 \boldsymbol{i} - \boldsymbol{j} q \quad \triangle$

what is vector A?

$\square + \triangle = 2 \boldsymbol{A} = 8 \boldsymbol{i} \implies \boldsymbol{A} = 4 \boldsymbol{i}$