Numerical Problem No. 2.10 (Vectors and Equilibrium) Physics HSSC - I (Class -11) (Punjab Curriculum and Text Board, Lahore) With Verified Answers

Given that $\to A$ $\vec {A}$ = $ˆ i$ $\hat {i}$ - 2 $ˆ j$ $\hat {j}$ + 3 $ˆ k$ $\hat {k}$ and $\to B$ $\vec {B}$ = 3 $ˆ i$ $\hat {i}$ - 4 $ˆ k$ $\hat {k}$ , find the projection of A on B. (Ans: -5/9 )

Given:

$\to A$ $\vec {A}$ = $ˆ i$ $\hat {i}$ - 2 $ˆ j$ $\hat {j}$ + 3 $ˆ k$ $\hat {k}$
$\to B$ $\vec {B}$ = 3 $ˆ i$ $\hat {i}$ - 4 $ˆ k$ $\hat {k}$

To Find:

The projection of vector $\to A$ $\vec {A}$ on $\to B$ $\vec {B}$ = A cos 𝞡 = ?

Solution:

We know that the projection of $\to A$ $\vec {A}$ is on $\to B$ $\vec {B}$ is A cos 𝜭, So, the scalar (dot) product is

$\to B$ $\vec {B}$ . $\to A$ $\vec {A}$ = B (A cos 𝜭)

or

A cos 𝜭 = $\frac{\to B . \to A}{B}$ $\frac {vec {B} . vec {A}}{B}$

A cos 𝜭 =

\frac{(3 ˆ i - 4 ˆ k) . (ˆ i - 2 ˆ j + 3 ˆ k)}{\sqrt{B x^{2} + B y^{2} + B z^{2}}}

$\frac {( 3\hat {i} - 4 \hat {k} ) . (\hat {i} - 2\hat {j} + 3\hat {k} )}{sqrt {Bx^2 + By^2 + Bz^2}}$

A cos 𝜭 =

\frac{(3 ｘ 1) ˆ i . ˆ i + (3 ｘ - 2) ˆ i . ˆ j + (3 ｘ 3) ˆ j . ˆ k + (- 4 ｘ 1) ˆ k . ˆ i + (- 4 ｘ - 2) ˆ k . ˆ j + (- 4 ｘ 3) ˆ k . ˆ k}{\sqrt{3^{2} + {(0)}^{2} + {(- 4)}^{2}}}

$\frac {(3 ｘ 1) hat {i} . hat {i} + (3 ｘ -2) hat {i} . hat {j} + (3 ｘ 3) hat {j} . hat {k} + (-4 ｘ 1) hat {k} . hat {i} + (-4 ｘ -2) hat {k} . hat {j} + (-4 ｘ 3) hat {k} . hat {k} }{sqrt {3^2 + (0)^2 + (-4)^2}}$

[∴ $ˆ i$ $hat {i}$ . $ˆ i$ $hat {i}$ = $ˆ j$ $hat {j}$ . $ˆ j$ $hat {j}$ = $ˆ k$ $hat {k}$ . $ˆ k$ $hat {k}$ =1
and
$ˆ i$ $hat {i}$ . $ˆ j$ $hat {j}$ = $ˆ j$ $hat {j}$ . $ˆ i$ $hat {i}$ = $ˆ j$ $hat {j}$ . $ˆ k$ $hat {k}$ = $ˆ k$ $hat {k}$ . $ˆ j$ $hat {j}$ = $ˆ k$ $hat {k}$ . $ˆ i$ $hat {i}$ = $ˆ i$ $hat {i}$ . $ˆ k$ $hat {k}$ = 0 ]

by putting the above values and calculating

A cos 𝜭 =

\frac{3 - 12}{\sqrt{9 + 16}}

$\frac {3 - 12}{sqrt {9 + 16}}$

A cos 𝜭 =

\frac{- 9}{\sqrt{25}}

$\frac {-9}{sqrt {25}}$

A cos 𝜭 =

\frac{- 9}{5}

$\frac {-9}{5}$ ------ Ans

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Numerical Problem 2.10
⇑

List of 11th Physics Numerical Problems of Chapter - 2

⇑

Numerical Problem 2.9 ⇚ 11th Physics Main Content List ⇛ Numerical Problem 2.11

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