The sides of parallelogram are 2 i ^ + 4 j ^ - 5 k ^ and i ^ + 2 j ^ + 3 k ^ . Find the unit vectors, parallel to their diagonals.

Here, D A → = a → = 2 i ^ + 4 j ^ - 5 k ^ , D B → = b → = i ^ + 2 j ^ + 3 k ^ By, parallelogram law of vector addition, D C → = D A → + D B → = a → + b → = 3 i ^ + 6 j ^ - 2 k ^ . Then D C ^ = 3 i ^ + 6 j ^ - 2 k ^ 3 2 + 6 2 + - 2 2 = 3 7 i ^ + 6 7 j ^ - 2 7 k ^ . Now, D B → + B A → = D A → ⇒ B A → = D A → - D B → = a → - b → = i ^ + 2 j ^ - 8 k ^ Then B A ^ = i ^ + 2 j ^ - 8 k ^ 1 2 + 2 2 + - 8 2 = 1 69 i ^ + 2 69 j ^ - 8 69 k ^ So, the required vectors parallel to the diagonals are 3 7 i ^ + 6 7 j ^ - 2 7 k ^ and 1 69 i ^ + 2 69 j ^ - 8 69 k ^ .

If G is the centroid of a triangle A B C , prove that G A → + G B → + G C → = 0 → .

For any triangle A B C , Given G A → + G B → + G C → = 0 → . Now G is the centroid of triangle A B C , which divides the medians A D , B E and C F in the ratio 2 : 1 ⇒ A G → : G D → = 2 : 1 ⇒ G A → = - 2 3 A D → Similarly, G B → = - 2 3 B E → and G C → = - 2 3 C F → Now, LHS = G A → + G B → + G C → = - 2 3 A D → + B E → + C F → = - 2 3 A B → + 1 2 B C → + B C → + 1 2 C A → + C A → + 1 2 A B → = - 2 3 3 2 ( A B → + B C → + C A → ) = - 1 0 → = 0 → = RHS .

(a) Show that the points a → - 2 b → + 3 c → ; 2 a → + 3 b → - 4 c → & - 7 b → + 10 c → are collinear.

Let A → = a → - 2 b → + 3 c → , B → = 2 a → + 3 b → - 4 c → and C → = - 7 b → + 10 c → . Here C → = 2 A → - B → ⇒ 2 A → - B → - C → = 0 → Thus points represented by A → , B → , C → are collinear.

E M B I B E

Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-4>Q 1

HARD

JEE Main/Advance

IMPORTANT

Earn 100

Let $A B C D$ be a tetrahedron such that the edges $A B, A C$ and $A D$ are mutually perpendicular. Let the area of triangles $A B C, A C D$ and $A D B$ be denoted by $x, y$ and $z$ sq. units respectively. Find the area of the triangle $B C D$ .

Important Questions on Vector Algebra

MEDIUM

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-4>Q 2

The sides of parallelogram are

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

and

\hat{i} + 2 \hat{j} + 3 \hat{k}

. Find the unit vectors, parallel to their diagonals.

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-4>Q 3

If

G

is the centroid of a triangle

A B C

, prove that

\vec{G A} + \vec{G B} + \vec{G C} = \vec{0}

.

EASY

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-4>Q 4

Find out whether the following pairs of lines are parallel, non-parallel & intersecting, or non-parallel & non-intersecting:
(a)

\vec{r_{1}} = \hat{i} + \hat{j} + 2 \hat{k} + λ (3 \hat{i} - 2 \hat{j} + 4 \hat{k})

,

\vec{r_{2}} = 2 \hat{i} + \hat{j} + 3 \hat{k} + μ (- 6 \hat{i} + 4 \hat{j} - 8 \hat{k})

.

MEDIUM

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-4>Q 5

Find out whether the following pairs of lines are parallel, non-parallel & intersecting, or non-parallel & non-intersecting:

(b) $\vec{r_{1}} = \hat{i} - \hat{j} + 3 \hat{k} + λ (\hat{i} - \hat{j} + \hat{k})$ , $\vec{r_{2}} = 2 \hat{i} + 4 \hat{j} + 6 \hat{k} + μ (2 \hat{i} + \hat{j} + 3 \hat{k})$

MEDIUM

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-4>Q 6

Find out whether the following pairs of lines are parallel, non-parallel & intersecting, or non-parallel & non-intersecting.

(c) $\vec{r_{1}} = \hat{i} + \hat{k} + λ (\hat{i} + 3 \hat{j} + 4 \hat{k})$ , $\vec{r_{2}} = 2 \hat{i} + 3 \hat{j} + μ (4 \hat{i} - \hat{j} + \hat{k})$