# Linear Dependence and Independence of Vectors

## Trending Questions

**Q.**In a triangle ABC, if |−−→BC|=3, |−−→CA|=5 and |−−→BA|=7, then the projection of the vector −−→BA on −−→BC is equal to:

- 132
- 192
- 152
- 112

**Q.**Let →a, →b and →c be three unit vectors, out of which vectors →b and →c are non-parallel. If α and β are the angles which vector →a makes with vectors →b and →c respectively and →a×(→b×→c)=12→b, then |α−β| is equal to :

- 60∘
- 45∘
- 30∘
- 90∘

**Q.**If →b and →c are two non-collinear unit vectors and →a is any vector, then (→a⋅→b)→b+(→a⋅→c)→c+→a⋅(→b×→c)|→b×→c|2(→b×→c)=

- →a
- →b
- →c
- →a+→b+→c

**Q.**

If →a, →b and →c are three vectors of equal magnitude. The angle between each pair of vectors is π3 such that ∣∣∣→a+→b+→c∣∣∣=√6. Then |→a| is equal to

- 2
- −1
- 1
√6

**Q.**The value of a for which the vectors

A=2i-j-4k ,

B=i-5j+ak and

C=2i+3j+4k are linearly dependent is

**Q.**

Answer the following as true or false.

→a and −→a are collinear- Data insufficient
- True
- False
- Ambiguous

**Q.**

If the vectors →a and →b are perpendicular to each other, then a vector →v in terms of →a and →b satisfying the equations

→v.→a=0, →v.→b=1 and [→v →a →b]=1 is

→b|→b|2+→a×→b|→a×→b|2

→b|→b|+→a×→b|→a×→b|2

→b|→b|2+→a×→b|→a×→b|

None of these

**Q.**In a triangle ABC, if |−−→BC|=3, |−−→CA|=5 and |−−→BA|=7, then the projection of the vector −−→BA on −−→BC is equal to:

- 112
- 192
- 132
- 152

**Q.**Vector →x is Vectors →x, →y and →z, each of magnitude √2, make an angle of 60∘ with each other.

→x×(→y×→z)=→a, →y×(→z×→x)=→b and →x×→y=→c.

- 12[(→a−→b)×→c+(→a+→b)]
- 12[(→a+→b)×→c+(→a−→b)]
- 12[−(→a+→b)×→c+(→a+→b)]
- 12[(→a+→b)×→c−(→a+→b)]

**Q.**Minimise Z=5x+4y Subject to constraints: 80x+100y>=88 40x+30y>=36 x, y>=0

**Q.**Prove that the following vectors are coplanar:

(i) $2\hat{i}-\hat{j}+\hat{k},\hat{i}-3\hat{j}-5\hat{k}\mathrm{and}3\hat{i}-4\hat{j}-4\hat{k}$

(ii) $\hat{i}+\hat{j}+\hat{k},2\hat{i}+3\hat{j}-\hat{k}\mathrm{and}-\hat{i}-2\hat{j}+2\hat{k}$

**Q.**Two vectors →a and →b are perpendicular to each other. A vector →c is inclined at an angle θ to both →a and →b. If |→a|=2, |→b|=3, |→c|=2 and →c=p→a+q→b+r(→a×→b), then

- p2=cos2θ
- r2=−cos2θ9
- r2=−cos2θ9
- q2=4cos2θ9

**Q.**Answer the following as true or false.

(i) →a and −→a are collinear

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

**Q.**In a triangle ABC, if |−−→BC|=3, |−−→CA|=5 and |−−→BA|=7, then the projection of the vector −−→BA on −−→BC is equal to:

- 132
- 152
- 112
- 192

**Q.**Find the values of a, b, c and d from the following equations:

$\left[\begin{array}{cc}2a+b& a-2b\\ 5c-d& 4c+3d\end{array}\right]=\left[\begin{array}{cc}4& -3\\ 11& 24\end{array}\right]$

**Q.**The solution of x

^{2}+ y

^{2}$\frac{dy}{dx}$ = 4, is

(a) x

^{2}+ y

^{2}= 12x + C

(b) x

^{2}+ y

^{2}= 3x + C

(c) x

^{3}+ y

^{3}= 3x + C

(d) x

^{3}+ y

^{3}= 12x + C

**Q.**If →a=^i+^j+^k, ^b=4^i+3^j+4^k and →c=^i+α^j+β^k are linearly dependent vectors and |→c|=√3, then

- α=1, β=−1
- α=1, β=±1
- α=−1, β=±1
- α=±1, β=1

**Q.**

Is the value of the determinant different along different r o w s and columns can we evaluate one's column in the area of the triangle and get the same answer?

**Q.**The top of a ladder 6 metres long is resting against a vertical wall on a level pavement, when the ladder begins to slide outwards. At the moment when the foot of the ladder is 4 metres from the wall, it is sliding away from the wall at the rate of 0.5 m/sec. How fast is the top-sliding downwards at this instance?

How far is the foot from the wall when it and the top are moving at the same rate?

**Q.**If ^a, ^b are unit vectors such that ^a+^b is also a unit vector then the angle between the vectors ^a and ^b is

- π6
- π4
- π3
- 2π3

**Q.**System of vectors a1, a2, .......an is said to be linearly dependent if there exists a system of scalars (c1, c2−−−cn such that c1¯a+c2¯a2+...cn¯an=¯0

- True
- False

**Q.**If →a=^i+^j+^k, →b=2^i−^j+^k and →c=^i+x^j+y^k, are linearly dependent and |→c|=√3 then (x, y) is

- (-1, -1)
- (2, 3)
- (3, 1)

**Q.**

If →a, →b, →c are three vectors such that |→a|=5, |→b|=12 and |→c|=13 and →a+→b+→c=0

Find the value of →a.→b+→b.→c+→c.→a.

**Q.**If $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ are non-zero, non-coplanar vectors, prove that the following vectors are coplanar:

(i) $5\overrightarrow{a}+6\overrightarrow{b}+7\overrightarrow{c,}7\overrightarrow{a}-8\overrightarrow{b}+9\overrightarrow{c}\mathrm{and}3\overrightarrow{a}+20\overrightarrow{b}+5\overrightarrow{c}$

(ii) $\overrightarrow{a}-2\overrightarrow{b}+3\overrightarrow{c},-3\overrightarrow{b}+5\overrightarrow{c}\mathrm{and}-2\overrightarrow{a}+3\overrightarrow{b}-4\overrightarrow{c}$

**Q.**The integral value of λ for which vectors λ2^i−2^j, 2λ2^j+64^k, −λ2^k−^i are linearly dependent, is equal to

**Q.**The complex number z1, z2 and the origin form an equilateral triangle only if z1z2+z2z1=1.

- True
- False

**Q.**Let →a, →b and →c be three unit vectors, out of which vectors →b and →c are non-parallel. If α and β are the angles which vector →a makes with vectors →b and →c respectively and →a×(→b×→c)=12→b, then |α−β| is equal to :

- 60∘
- 45∘
- 30∘
- 90∘

**Q.**Two non zero vectors →v1 and →v2 which are orthogonal to each other will always be linearly

- dependent
- independent
- either dependent or independent

**Q.**Let ¯a=2¯i−3¯j+^k, ¯b=^i−^j+3^k and ¯c=^i+^j−^k Then the system of vectors ¯a, ¯band¯c

- Cannot be determined
- None of the above
- Is linearly dependent
- Is linearly independent

**Q.**Two non zero vectors →v1 and →v2 which are orthogonal to each other will always be linearly

- dependent
- independent
- either dependent or independent