QBManager

Vector Algebra

87761 If \(\vec{a}=2 \hat{i}+\hat{j}-8 \hat{k}\) and \(\vec{b}=\hat{i}+3 \hat{j}-4 \hat{k}\) then the magnitude of \(\vec{a}+\vec{b}\) is equal to

1 13

2 \(\frac{13}{3}\)

3 \(\frac{3}{13}\)

4 \(\frac{4}{13}\)

AMU-2002

Vector Algebra

87762 If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three non-coplanar vectors such that \(\vec{a}+\vec{b}+\vec{c}=\alpha \vec{d}\) and \(\vec{b}+\vec{c}+\vec{d}=\beta \vec{a}\) then what is \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{d}}\) equal to?

1 \(\overrightarrow{0}\)

2 \(\alpha \vec{a}\)

3 \(\beta \overrightarrow{\mathrm{b}}\)

4 \((\alpha+\beta) \overrightarrow{\mathrm{c}}\)

Explanation:

(A) : \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}\) and \(\overrightarrow{\mathrm{c}}\) are threenon - coplanar
\(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{d}}\)
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=\beta \overrightarrow{\mathrm{a}}\)
\(\overrightarrow{\mathrm{d}}=\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{\alpha},\)
\(\overrightarrow{\mathrm{a}}=\frac{\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}}{\beta}\)
Now,
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=\beta \overrightarrow{\mathrm{a}}\)
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{\alpha}=\beta \overrightarrow{\mathrm{a}}\)
\(\frac{\alpha \overrightarrow{\mathrm{b}}+\alpha \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{\alpha}=\beta \overrightarrow{\mathrm{a}}\)
\((\alpha+1) \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{a}}+(\alpha+1) \overrightarrow{\mathrm{c}}=\alpha \beta \overrightarrow{\mathrm{a}}+0 \overrightarrow{\mathrm{b}}+0 \overrightarrow{\mathrm{c}}\)
Coefficient comparison
\(\alpha+1=0 \quad \Rightarrow \alpha=-1\)
\(1=\alpha \beta\)
\(1=(-1) \beta\)
\(\beta=-1\)
From (i) we get -
\(\alpha=-1 \Rightarrow \vec{a}+\vec{b}+\vec{c}=-\vec{d} \Rightarrow \vec{a}+\vec{b}+\vec{c}+\vec{d}=\overrightarrow{0}\)
From (ii) we get -
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=(-1) \overrightarrow{\mathrm{a}}\)
\(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=\overrightarrow{0}\)

SCRA-2009

Vector Algebra

87764 Consider the following statements:
(i). The sum of two unit vectors can be a unit vector.
(ii). The magnitude of the difference between two unit vectors can be greater than the magnitude of a unit vector.
Which of the above statements is/are correct?

1 1 only

2 2 only

3 Both 1 and 2

4 Neither 1 nor 2

SCRA-2012

Vector Algebra

87765 Let \(\overrightarrow{\mathrm{b}}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) and \(\overrightarrow{\mathrm{c}}\) be two vectors perpendicular to each other in xy-plane, then a vector in the same plane having projections 1 and 2 along \(\vec{b}\) and \(\overrightarrow{\mathbf{c}}\), respectively, is

1 \(\hat{i}+2 \hat{j}\)

2 \(2 \hat{i}-\hat{j}\)

3 \(2 \hat{i}+\hat{j}\)

4 none of these

AMU-2010

Vector Algebra

87761 If \(\vec{a}=2 \hat{i}+\hat{j}-8 \hat{k}\) and \(\vec{b}=\hat{i}+3 \hat{j}-4 \hat{k}\) then the magnitude of \(\vec{a}+\vec{b}\) is equal to

1 13

2 \(\frac{13}{3}\)

3 \(\frac{3}{13}\)

4 \(\frac{4}{13}\)

AMU-2002

Vector Algebra

87762 If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three non-coplanar vectors such that \(\vec{a}+\vec{b}+\vec{c}=\alpha \vec{d}\) and \(\vec{b}+\vec{c}+\vec{d}=\beta \vec{a}\) then what is \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{d}}\) equal to?

1 \(\overrightarrow{0}\)

2 \(\alpha \vec{a}\)

3 \(\beta \overrightarrow{\mathrm{b}}\)

4 \((\alpha+\beta) \overrightarrow{\mathrm{c}}\)

Explanation:

(A) : \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}\) and \(\overrightarrow{\mathrm{c}}\) are threenon - coplanar
\(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{d}}\)
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=\beta \overrightarrow{\mathrm{a}}\)
\(\overrightarrow{\mathrm{d}}=\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{\alpha},\)
\(\overrightarrow{\mathrm{a}}=\frac{\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}}{\beta}\)
Now,
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=\beta \overrightarrow{\mathrm{a}}\)
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{\alpha}=\beta \overrightarrow{\mathrm{a}}\)
\(\frac{\alpha \overrightarrow{\mathrm{b}}+\alpha \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{\alpha}=\beta \overrightarrow{\mathrm{a}}\)
\((\alpha+1) \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{a}}+(\alpha+1) \overrightarrow{\mathrm{c}}=\alpha \beta \overrightarrow{\mathrm{a}}+0 \overrightarrow{\mathrm{b}}+0 \overrightarrow{\mathrm{c}}\)
Coefficient comparison
\(\alpha+1=0 \quad \Rightarrow \alpha=-1\)
\(1=\alpha \beta\)
\(1=(-1) \beta\)
\(\beta=-1\)
From (i) we get -
\(\alpha=-1 \Rightarrow \vec{a}+\vec{b}+\vec{c}=-\vec{d} \Rightarrow \vec{a}+\vec{b}+\vec{c}+\vec{d}=\overrightarrow{0}\)
From (ii) we get -
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=(-1) \overrightarrow{\mathrm{a}}\)
\(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=\overrightarrow{0}\)

SCRA-2009

Vector Algebra

87764 Consider the following statements:
(i). The sum of two unit vectors can be a unit vector.
(ii). The magnitude of the difference between two unit vectors can be greater than the magnitude of a unit vector.
Which of the above statements is/are correct?

1 1 only

2 2 only

3 Both 1 and 2

4 Neither 1 nor 2

SCRA-2012

Vector Algebra

87765 Let \(\overrightarrow{\mathrm{b}}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) and \(\overrightarrow{\mathrm{c}}\) be two vectors perpendicular to each other in xy-plane, then a vector in the same plane having projections 1 and 2 along \(\vec{b}\) and \(\overrightarrow{\mathbf{c}}\), respectively, is

1 \(\hat{i}+2 \hat{j}\)

2 \(2 \hat{i}-\hat{j}\)

3 \(2 \hat{i}+\hat{j}\)

4 none of these

AMU-2010

Vector Algebra

87761 If \(\vec{a}=2 \hat{i}+\hat{j}-8 \hat{k}\) and \(\vec{b}=\hat{i}+3 \hat{j}-4 \hat{k}\) then the magnitude of \(\vec{a}+\vec{b}\) is equal to

1 13

2 \(\frac{13}{3}\)

3 \(\frac{3}{13}\)

4 \(\frac{4}{13}\)

AMU-2002

Vector Algebra

87762 If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three non-coplanar vectors such that \(\vec{a}+\vec{b}+\vec{c}=\alpha \vec{d}\) and \(\vec{b}+\vec{c}+\vec{d}=\beta \vec{a}\) then what is \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{d}}\) equal to?

1 \(\overrightarrow{0}\)

2 \(\alpha \vec{a}\)

3 \(\beta \overrightarrow{\mathrm{b}}\)

4 \((\alpha+\beta) \overrightarrow{\mathrm{c}}\)

Explanation:

(A) : \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}\) and \(\overrightarrow{\mathrm{c}}\) are threenon - coplanar
\(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{d}}\)
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=\beta \overrightarrow{\mathrm{a}}\)
\(\overrightarrow{\mathrm{d}}=\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{\alpha},\)
\(\overrightarrow{\mathrm{a}}=\frac{\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}}{\beta}\)
Now,
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=\beta \overrightarrow{\mathrm{a}}\)
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{\alpha}=\beta \overrightarrow{\mathrm{a}}\)
\(\frac{\alpha \overrightarrow{\mathrm{b}}+\alpha \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{\alpha}=\beta \overrightarrow{\mathrm{a}}\)
\((\alpha+1) \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{a}}+(\alpha+1) \overrightarrow{\mathrm{c}}=\alpha \beta \overrightarrow{\mathrm{a}}+0 \overrightarrow{\mathrm{b}}+0 \overrightarrow{\mathrm{c}}\)
Coefficient comparison
\(\alpha+1=0 \quad \Rightarrow \alpha=-1\)
\(1=\alpha \beta\)
\(1=(-1) \beta\)
\(\beta=-1\)
From (i) we get -
\(\alpha=-1 \Rightarrow \vec{a}+\vec{b}+\vec{c}=-\vec{d} \Rightarrow \vec{a}+\vec{b}+\vec{c}+\vec{d}=\overrightarrow{0}\)
From (ii) we get -
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=(-1) \overrightarrow{\mathrm{a}}\)
\(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=\overrightarrow{0}\)

SCRA-2009

Vector Algebra

87764 Consider the following statements:
(i). The sum of two unit vectors can be a unit vector.
(ii). The magnitude of the difference between two unit vectors can be greater than the magnitude of a unit vector.
Which of the above statements is/are correct?

1 1 only

2 2 only

3 Both 1 and 2

4 Neither 1 nor 2

SCRA-2012

Vector Algebra

87765 Let \(\overrightarrow{\mathrm{b}}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) and \(\overrightarrow{\mathrm{c}}\) be two vectors perpendicular to each other in xy-plane, then a vector in the same plane having projections 1 and 2 along \(\vec{b}\) and \(\overrightarrow{\mathbf{c}}\), respectively, is

1 \(\hat{i}+2 \hat{j}\)

2 \(2 \hat{i}-\hat{j}\)

3 \(2 \hat{i}+\hat{j}\)

4 none of these

AMU-2010

Vector Algebra

87761 If \(\vec{a}=2 \hat{i}+\hat{j}-8 \hat{k}\) and \(\vec{b}=\hat{i}+3 \hat{j}-4 \hat{k}\) then the magnitude of \(\vec{a}+\vec{b}\) is equal to

1 13

2 \(\frac{13}{3}\)

3 \(\frac{3}{13}\)

4 \(\frac{4}{13}\)

AMU-2002

Vector Algebra

87762 If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three non-coplanar vectors such that \(\vec{a}+\vec{b}+\vec{c}=\alpha \vec{d}\) and \(\vec{b}+\vec{c}+\vec{d}=\beta \vec{a}\) then what is \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{d}}\) equal to?

1 \(\overrightarrow{0}\)

2 \(\alpha \vec{a}\)

3 \(\beta \overrightarrow{\mathrm{b}}\)

4 \((\alpha+\beta) \overrightarrow{\mathrm{c}}\)

Explanation:

(A) : \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}\) and \(\overrightarrow{\mathrm{c}}\) are threenon - coplanar
\(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{d}}\)
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=\beta \overrightarrow{\mathrm{a}}\)
\(\overrightarrow{\mathrm{d}}=\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{\alpha},\)
\(\overrightarrow{\mathrm{a}}=\frac{\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}}{\beta}\)
Now,
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=\beta \overrightarrow{\mathrm{a}}\)
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{\alpha}=\beta \overrightarrow{\mathrm{a}}\)
\(\frac{\alpha \overrightarrow{\mathrm{b}}+\alpha \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{\alpha}=\beta \overrightarrow{\mathrm{a}}\)
\((\alpha+1) \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{a}}+(\alpha+1) \overrightarrow{\mathrm{c}}=\alpha \beta \overrightarrow{\mathrm{a}}+0 \overrightarrow{\mathrm{b}}+0 \overrightarrow{\mathrm{c}}\)
Coefficient comparison
\(\alpha+1=0 \quad \Rightarrow \alpha=-1\)
\(1=\alpha \beta\)
\(1=(-1) \beta\)
\(\beta=-1\)
From (i) we get -
\(\alpha=-1 \Rightarrow \vec{a}+\vec{b}+\vec{c}=-\vec{d} \Rightarrow \vec{a}+\vec{b}+\vec{c}+\vec{d}=\overrightarrow{0}\)
From (ii) we get -
\(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=(-1) \overrightarrow{\mathrm{a}}\)
\(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=\overrightarrow{0}\)

SCRA-2009

Vector Algebra

87764 Consider the following statements:
(i). The sum of two unit vectors can be a unit vector.
(ii). The magnitude of the difference between two unit vectors can be greater than the magnitude of a unit vector.
Which of the above statements is/are correct?

1 1 only

2 2 only

3 Both 1 and 2

4 Neither 1 nor 2

SCRA-2012

Vector Algebra

87765 Let \(\overrightarrow{\mathrm{b}}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) and \(\overrightarrow{\mathrm{c}}\) be two vectors perpendicular to each other in xy-plane, then a vector in the same plane having projections 1 and 2 along \(\vec{b}\) and \(\overrightarrow{\mathbf{c}}\), respectively, is

1 \(\hat{i}+2 \hat{j}\)

2 \(2 \hat{i}-\hat{j}\)

3 \(2 \hat{i}+\hat{j}\)

4 none of these

AMU-2010