Question

f three sides of a parallelopiped are formed by

A = î + 2 ĵ + k, B = î – ĵ + 2k and C = 2î + ĵ – k,
volume?
​

Answer 1

ANSWER :-

➞ 14 cubic units.

EXPLANATION :-

We are given three sides of a parallelepiped, as vectors,

• A = i + 2j + k
• B = i - j + 2k
• C = 2i + j - k

We have to find the volume of the given Parallelepiped.

For a parallelepiped, the volume is given by,

⇒ Volume = A . (B × C)

So, Let us find B × C, first

| i j k |

⇒ B × C = | 1 -1 2 |

| 2 1 -1 |

⇒ B × C = (-1×-1 - 2×1)i - (2×-1 - 2×2)j + (1×1 - 2×-1)k

⇒ B × C = (1 - 2)i - (-2 - 4)j + (1 + 2)k

⇒ B × C = -i + 6j + 3k

Now, we have

• A = i + 2j + k
• B × C = -i + 6j + 3k

So,

⇒ A . (B × C)

⇒ (i + 2j + k) . (-i + 6j + 3k)

⇒ (-1×1 + 2×6 + 3×1)

⇒ -1 + 12 + 3

⇒ 14 cubic units.

Answer 2

☃️Here we go !!

ANSWER :

Option c)

$\sf -4 \hat{i} \: - \: 2 \hat{j} \: + \: 5 \hat{k}$

GIVEN :

$\sf \hat{i} \: - 3 \hat{j} \: + 2 \hat{k}$

$\sf 3 \hat{i} \: + 6 \hat{j} \: - 7 \hat{k}$

TO FIND :

The vector that must be added to the given vector so that the resultant vector is a unit vector along y - axis.

SOLUTION :

$\implies \sf A + B + C \: = \: j$

Here,

→ J is the unit vector.

→ C is the resultant vector.

$\implies \sf B - A \: = \: (\hat{i} \: - 3 \hat{j} \: + \: 2 \hat{k}) \: - \: (\hat{i} \: + \: 6 \hat{j} \: - 7 \hat{k})</p><p>$

$\implies \sf A + B \: = \: 4 \hat{i} \: + \: 3 \hat{j} \: - \: 5 \hat{k}</p><p>$

$\sf Now, \: Using \: subst \: rule,$

$\implies \sf 4 \hat{i} \: + \: 3 \hat{j} \: + \: c \: = \: j$

$\implies \sf c \: = \: j \: - \: (4 \hat{i} \: + \: 3 \hat{j} \: - \: 5 \hat{k})$

$\implies \sf j \: = \: 4 \hat{i} \: - \: 3 \hat{j} \: + \: 5 \hat{k}</p><p>$

$\implies \sf -4 \hat{i} \: - \: 2 \hat{j} \: + \: 5 \hat{k}</p><p>$

$\therefore \sf -4 \hat{i} \: - \: 2 \hat{j} \: + \: 5 \hat{k}$

☃️Hence we are done !!