If A ( 2 , 1 , 3 ) , B ( 2 , - 3 , 4 ) and C ( - 1 , 2 , 7 ) , find a point D such that A B → and C D → are parallel.

Let the coordinates of point D be ( α , β , γ ) . Since, A B → and C D → are parallel. A B → = λ C D → for some scalar λ ≠ 0 . . . ( 1 ) Here P.V of A = 2 i + j + 3 k ; P.V of B = 2 i - 3 j + 4 k P.V of C = - i + 2 j + 7 k A B → = P . V of B - P . V of A = ( 2 i - 3 j + 4 k ) - ( 2 i + j + 3 k ) = - 4 j + k C D → = P . V of D - P . V of C = ( α i + β j + γ k ) - ( - i + 2 j + 7 k ) = ( α + 1 ) i + ( β - 2 ) j + ( γ - 7 ) k From ( 1 ) ; we have - 4 j + k = λ ( α + 1 ) i + ( β - 2 ) j + ( γ - 7 ) k 0 = λ ( α + 1 ) . . . ( 2 ) - 4 = λ ( β - 2 ) . . . ( 3 ) and 1 = λ ( γ - 7 ) . . . ( 4 ) Solving ( 2 ) , ( 3 ) , ( 4 ) α = - 1 , β = 2 - 4 λ , γ = 1 λ + 7 On taking λ = 1 ; α = - 1 ; β = 2 - 4 = - 2 and γ = 1 + 7 = 8 Hence, the required point D b e D ( - 1 , - 2 , 8 ) . For different values of λ , we get different points at which A B → a n d C D → are parallel.

If A ( 2 , 1 , 3 ) , B ( 2 , - 3 , 4 ) and C ( - 1 , 2 , 7 ) , find a point D such that A B → and C D → are parallel.

Let the coordinates of point D be ( α , β , γ ) . Since, A B → and C D → are parallel. A B → = λ C D → for some scalar λ ≠ 0 . . . ( 1 ) Here P.V of A = 2 i + j + 3 k ; P.V of B = 2 i - 3 j + 4 k P.V of C = - i + 2 j + 7 k A B → = P . V of B - P . V of A = ( 2 i - 3 j + 4 k ) - ( 2 i + j + 3 k ) = - 4 j + k C D → = P . V of D - P . V of C = ( α i + β j + γ k ) - ( - i + 2 j + 7 k ) = ( α + 1 ) i + ( β - 2 ) j + ( γ - 7 ) k From ( 1 ) ; we have - 4 j + k = λ ( α + 1 ) i + ( β - 2 ) j + ( γ - 7 ) k 0 = λ ( α + 1 ) . . . ( 2 ) - 4 = λ ( β - 2 ) . . . ( 3 ) and 1 = λ ( γ - 7 ) . . . ( 4 ) Solving ( 2 ) , ( 3 ) , ( 4 ) α = - 1 , β = 2 - 4 λ , γ = 1 λ + 7 On taking λ = 1 ; α = - 1 ; β = 2 - 4 = - 2 and γ = 1 + 7 = 8 Hence, the required point D b e D ( - 1 , - 2 , 8 ) . For different values of λ , we get different points at which A B → a n d C D → are parallel.

Find a unit vector in the direction of a → - b → , where a → = i ^ + j ^ + k ^ and b → = 2 i ^ + j ^ - 3 k ^

Given a → = - i + j + k ; b → = 2 i + j - 3 k a → - b → = - i + j + k - 2 i + j - 3 k = - 3 i + 4 k and | a → - b → | = ( - 3 ) 2 + 4 2 = 9 + 16 = 5 Unit vector in the direction of a → - b → = a → - b → a → - b → = - 3 i + 4 k 5

If a → = i ^ + j ^ , b → = j ^ + k ^ and c → = i ^ + k ^ , find a unit vector in the direction of a → - 2 b → + 3 c → .

Given a → = i + j ; b → = j + k and c → = i + k a → - 2 b → + 3 c → = i + j - 2 ( j + k ) + 3 ( i + k ) = 4 i - j + k | a → - 2 b → + 3 c → | = 4 2 + ( - 1 ) 2 + 1 2 = 18 = 3 2 Thus, required unit vector in the direction of a → - 2 b → + 3 c → = a → - 2 b → + 3 c → | a → - 2 b → + 3 c → | = ( 4 i - j + k ) 3 2

If the vectors 4 i ^ + α j ^ + k ^ and β i ^ + 2 j ^ + k ^ are parallel, find the values of α and β .

Let a → = 4 i + α j + k b → = β i + 2 j + k Since a → and b → are parallel. a → = λ b → for some scalar λ ≠ 0 . ⇒ ( 4 i + α i + k ) = λ ( β i + 2 j + k ) ⇒ 4 = λ β ; α = 2 λ ; 1 = λ α = 2 × 1 = 2 ; β = 4 1 = 4 Hence, α = 2 , β = 4 .

If a → = b → + c → , then is it true that a → = b → + c → ? Justify your answer.

Let b → = 4 i ^ ; c → = 3 j ^ a → = b → + c → = 4 i ^ + 3 j ^ | a → | = 4 2 + 3 2 = 16 + 9 = 5 Here | b → | = | 4 i ^ | = 4 ; | c → | = | 3 j ^ | = 3 i.e. | a → | ≠ | b → | + | c → | So if a → = b → + c → Then it is not true always that | a → | = | b → | + | c → |

E M B I B E

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>CHAPTER TEST>Q 9

MEDIUM

12th ICSE

IMPORTANT

Earn 100

If $A (2, 1, 3), B (2, - 3, 4) and C (- 1, 2, 7)$ , find a point $D$ such that $\vec{A B}$ and $\vec{C D}$ are parallel.

Important Questions on Vectors

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>CHAPTER TEST>Q 10

Find a unit vector in the direction of $\vec{a} - \vec{b}$ , where $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = 2 \hat{i} + \hat{j} - 3 \hat{k}$

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>CHAPTER TEST>Q 10

If $\vec{a} = \hat{i} + \hat{j}, \vec{b} = \hat{j} + \hat{k}$ and $\vec{c} = \hat{i} + \hat{k}$ , find a unit vector in the direction of $\vec{a} - 2 \vec{b} + 3 \vec{c}$ .

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>CHAPTER TEST>Q 11

If the vectors

4 \hat{i} + α \hat{j} + \hat{k}

and

β \hat{i} + 2 \hat{j} + \hat{k}

are parallel, find the values of

α

and

β

.

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>CHAPTER TEST>Q 12

If

\vec{a} = \vec{b} + \vec{c}

, then is it true that

|\vec{a}| = |\vec{b}| + |\vec{c}|

? Justify your answer.

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>CHAPTER TEST>Q 13

If $\vec{a} = \hat{i} - \hat{j}$ and $\vec{b} = - \hat{j} + 2 \hat{k}$ , then find $(\vec{a} + \vec{b}) \cdot (\vec{a} - 2 \vec{b})$ .

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>CHAPTER TEST>Q 14

Show that the direction cosines of a vector equally inclined to the axes $O X, O Y$ and $O Z$ are $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>CHAPTER TEST>Q 14

A unit vector

\vec{a}

makes angles

\frac{π}{4}

and

\frac{π}{3}

with x-axis and y-axis respectively and an acute angle

θ

with z-axis, then find

θ

and the (scalar and vector) components of

\vec{a}

along the axes.

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>CHAPTER TEST>Q 15

If $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular vectors of equal magnitude, show that the vector $\vec{a} + \vec{b} + \vec{c}$ is equally inclined to $\vec{a}, \vec{b}$ and $\vec{c}$ .