A vector A is added to B=6i-8j. The resultant vector is in the positive x direction and has a magnitude equal to A . What is the

Question

ale4655 [162]

2 years ago

12

A vector A is added to B=6i-8j. The resultant vector is in the positive x direction and has a magnitude equal to A . What is the

magnitude of A?
a)11
b)7.1
c)5.1
d)8.3
e)12.2

Physics

2 answers:

snow_tiger [21] · Answer 1 · 2023-04-16T00:12:05+03:00

The correct answer would be letter d, 8.3.

Solution for the problem follows:

Given are:

B = 6i - 8j

A is unknown; let A be = mi + nj

A+B is along the x axis (therefore A+B = Ki + 0j, where K is unknown,

but then again the magnitude of A+B is the similar as the magnitude of A,

so mag(A+B)=K=sqrt(m^2+n^2), or K^2 = m^2+n^2.

A+B, from simple vector addition, will be now (m+6)i + (n-8)j.

Ever since we previously know A+B = Ki + 0j, we now know that:

m+6 = K

n-8 = 0, which implies n=8.

Thus, K^2=m^2+n^2 ====> (m+6)^2 = m^2 +8^2

= m^2 + 12m + 36 = m^2 + 64

= 12m = 28

= m = 2.33333...

Therefore, the magnitude of A is sqrt[(2.333...)^2 + 8^2] = 8.3333

Eva8 [605] · Answer 2 · 2023-04-16T01:52:54+03:00

<u>The magnitude of $\vec A$ is $8.3$ .</u>

Further explanation:

A vector is a quantity having magnitude and direction both. It is represented as the product of magnitude and direction vector.

Given:

The vector is given as $\vec B = 6\hat i - 8\hat j$ .

The direction of resultant vector is in the X- direction.

Concept used:

Consider a vector $\vec A = a\hat i + b\hat j$ which is added to $\vec B$ to get a resultant vector $\vec C$ . The resultant vector is directed in the positive X-direction which means that the y- component of the vector is zero.

The expression for the resultant vector is given as.

$\vec C = \vec A + \vec B$

Substitute $6\hat i - 8\hat j$ for $\vec B$ in the above expression.

$\begin{gathered}\vec C = \left( {a\hat i + b\hat j} \right) + \left( {6\hat i - 8\hat j} \right) \\= \left( {a + 6} \right)\hat i + \left( {b - 8} \right)\hat j \\ \end{gathered}$

The resultant vector is represented as.

$\vec C = x\hat i + 0\hat j$

Compare the above two expression of resultant vector.

$x = a + 6$ …… (1)

$\begin{aligned}0&=b-8\\b&=8\\\end{aligned}$

The magnitude of resultant vector is equal to the magnitude of $\vec A$ .

The expression for the magnitude of $\vec A$ is given as.

$\left| {\vec A} \right| = \sqrt {\left( {{a^2}} \right) + \left( {{b^2}} \right)}$ …… (2)

The expression for the magnitude of $\vec C$ is given as.

$\left| {\vec C} \right| = \sqrt {{{\left( {a + 6} \right)}^2}}$

Compare the above two expressions.

$\sqrt {\left( {{a^2}} \right) + \left( {{b^2}} \right)}= \sqrt {{{\left( {a + 6} \right)}^2}}$

Substitute 8 for $b$ in above expression.

$\begin{aligned}\sqrt {\left( {{a^2}} \right) + \left( {{8^2}} \right)}&=\sqrt {{{\left( {a + 6} \right)}^2}}\\{a^2} + 64&={a^2} + 36 + 12a \\12a&=28 \\a&=2.33 \\ \end{aligned}$