Vector Algebra Test

Vector Algebra

This is Vector Algebra Test-03 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
Magnitude of the vector \(\vec a = \frac{1}{{\sqrt 3 }}\hat i + \frac{1}{{\sqrt 3 }}\hat j + \frac{1}{{\sqrt 3 }}\hat k\) is
  • A
    0.5
  • B
    1
    Correct
  • C
    \(\sqrt 3 \)
  • D
    1.5
2
Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (– 5, 7).
  • A
    –7 and 6; \( - 7\hat i\;and\;6\hat j\)
    Correct
  • B
    –7 and -6; \( - 7\hat i\;and - 6\hat j\)
  • C
    –7 and 6; \(7\hat i\;and - 6\hat j\)
  • D
    7 and 6; \(7\hat i\;and\;6\hat j\)
3
Find the values of x and y so that the vectors \(2\hat i + 3\hat j\;and\;x\hat i + y\hat j\) are equal
  • A
    x = 2, y = 2
  • B
    x = 2, y = 3
    Correct
  • C
    x = 3, y = 3
  • D
    x = 3, y = 2
4
Find the sum of the vectors\(\vec a = \hat i - 2\hat j + \hat k,\;\vec b = - 2\hat i + 4\hat j + 5\hat k\;\) and \(\vec c = \hat i - 6\hat j - 7\hat k\)
  • A
    \( - 4\hat j - \hat k\)
    Correct
  • B
    \(\hat i - 4\hat j - \hat k\)
  • C
    \( - \hat i - 4\hat j - \hat k\)
  • D
    \( - \hat i + 4\hat j - \hat k\)
5
Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.
  • A
    \(\frac{{\sqrt 3 }}{2}\hat i + \frac{1}{2}\hat j\)
    Correct
  • B
    \( - \frac{{\sqrt 3 }}{2}\hat i - \frac{1}{2}\hat j\)
  • C
    \(\frac{{\sqrt 3 }}{2}\hat i - \frac{1}{2}\hat j\)
  • D
    \( - \frac{{\sqrt 3 }}{2}\hat i + \frac{1}{2}\hat j\)
6
Find the value of x for which \(x\left( {\hat i + \hat j + \hat k} \right)\)is a unit vector
  • A
    \( \pm \frac{1}{{\sqrt 7 }}\)
  • B
    \( \pm \frac{1}{{\sqrt 3 }}\)
    Correct
  • C
    \( \pm \frac{1}{{\sqrt 2 }}\)
  • D
    \( \pm \frac{1}{{\sqrt 5 }}\)
7
Show that the points A(1, – 2, – 8), B (5, 0, – 2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.
  • A
    2 : 3
    Correct
  • B
    2 :1
  • C
    2 :4
  • D
    3 :2
8
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are \(\left( {2\vec a + \vec b} \right)\)and\(\left( {\vec a - 3\vec b} \right)\) externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ.
  • A
    \(5\vec a + 5\vec b\)
  • B
    \(5\vec a + 3\vec b\)
  • C
    \(3\vec a + 5\vec b\)
    Correct
  • D
    \(3\vec a + 3\vec b\)
9
Let \(\vec a = \hat i + 4\hat j + 2\hat k\;,\;\;\vec b = 3\hat i - 2\hat j + 7\hat k \)\(and\;\vec c = 2\hat i - \hat j + 4\hat k\;\) . Find a vector \(\vec d\) which is perpendicular to both \(\vec a\;and\;\vec b,\;and\;\vec c.\vec d = 15.\)
  • A
    \(\frac{1}{3}\left( { - 160\hat i - 5\hat j + 70\hat k} \right)\)
  • B
    \(\frac{1}{3}\left( {160\hat i - 5\hat j - 70\hat k} \right)\)
  • C
    \(\frac{1}{3}\left( {160\hat i - 5\hat j + 70\hat k} \right)\)
    Correct
  • D
    \(\frac{1}{3}\left( {160\hat i + 5\hat j - 70\hat k} \right)\)
10
The scalar product of the vector \(\hat i + \hat j + \hat k\) with a unit vector along the sum of vectors \(2\hat i + 4\hat j - 5\hat k\;and\;\lambda \hat i + 2\hat j + 3\hat k\) is equal to one. Find the value of\(\;\lambda \).
  • A
    \(\lambda \) = 1
    Correct
  • B
    \(\lambda \) = 2
  • C
    \(\lambda \) = -1
  • D
    \(\lambda \) = -2
11
The scalar product of two nonzero vectors \(\vec a\;and\;\vec b\;\) is denoted
  • A
    \(\vec a.\vec b\)
    Correct
  • B
    \(\overleftrightarrow {ab}\)
  • C
    $ab$
  • D
    \(\overrightarrow {ab} \)
12
The scalar product of two nonzero vectors \(\vec a\;and\;\vec b\;\) is defined as
  • A
    \(\vec a.\vec b = 2\left| {\vec a} \right|\left| {\vec b} \right|\sin \theta \)
  • B
    \(\vec a.\vec b = \left| {\vec a} \right|\left| {\vec b} \right|\sin \theta \)
  • C
    \(\vec a.\vec b = 2\left| {\vec a} \right|\left| {\vec b} \right|\cos \theta \)
  • D
    \(\vec a.\vec b = \left| {\vec a} \right|\left| {\vec b} \right|\cos \theta \)
    Correct
13
If \(\vec a = {a_1}\hat i + {a_2}\hat j + {a_3}\hat k\;\) and \(\vec b = {b_1}\hat i + {b_2}\hat j + {b_3}\hat k\) then the dot product \(\vec a.\vec b = \)
  • A
    \({a_1}{b_1} + {a_2}{b_2} - {a_3}{b_3}\)
  • B
    \({a_1}{b_1} - {a_2}{b_2} - {a_3}{b_3}\)
  • C
    \({a_1}{b_1} + {a_2}{b_2} + {a_3}{b_3}\)
    Correct
  • D
    \({a_1}{b_1} - {a_2}{b_2} + {a_3}{b_3}\)
14
If \(\theta \) is the angle between vectors \(\vec a\;and\;\vec b\) then the cross product \(\vec a \times \vec b = \)
  • A
    \(\left| a \right|\left| b \right|\cos \theta \)
  • B
    \(\left| a \right|\left| b \right|\sin \theta \hat n\)
    Correct
  • C
    \(\left| a \right|\left| b \right|\sin \theta \)
  • D
    \(2\left| a \right|\left| b \right|\sin \theta \hat n\)
15
Find the unit vector in the direction of the vector\(\vec a = \hat i + \hat j + 2\hat k\)
  • A
    \(\vec a = \frac{1}{{\sqrt 6 }}\hat i + \frac{1}{{\sqrt 6 }}\hat j + \frac{2}{{\sqrt 6 }}\hat k\)
    Correct
  • B
    \(\vec a = \frac{1}{{\sqrt 6 }}\hat i - \frac{1}{{\sqrt 6 }}\hat j + \frac{2}{{\sqrt 6 }}\hat k\)
  • C
    \(\vec a = - \frac{1}{{\sqrt 6 }}\hat i + \frac{1}{{\sqrt 6 }}\hat j + \frac{2}{{\sqrt 6 }}\hat k\)
  • D
    \(\vec a = \frac{1}{{\sqrt 6 }}\hat i + \frac{1}{{\sqrt 6 }}\hat j - \frac{2}{{\sqrt 6 }}\hat k\)