CBSE Class 12th Mathematics Chapter 10 - Vectors Important Questions with Answers

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If the sum of two unit vectors a and b is a unit vector, show that the magnitude of their difference is ?3. (Delhi 2019, 2012c)

Question 2.

If \(\vec{a}\) = 2i? + 5j? + k?, \(\vec{b}\) = i? – 2 j? + k? and \(\vec{c}\) = – 3i? + j? + 2k?, find \([\vec{a} \vec{b} \vec{c}]\). (Delhi 2019)

Question 3.

If |\(\vec{a}\)| = 2, |\(\vec{b}\)| = 7 and \(\vec{a} \times \vec{b}\) = 3i? + 2j? + 6k?, find the angle between \(\vec{a}\) and \(\vec{b}\). (All India 2019)

Question 4.

Find the volume of cuboid whose edges are given by -5i? + 7j? + 5k?, -5i? + 7j? – 5k? and 7i? – 5 j? – 5k?. (All India 2019)

Question 5.

Show that the points A(-2i? + 5j? + 5k?), B(i? + 2 j? + 5k?) and C(7i? – k?) are collinear. (All India 2019)

Question 6.

Find \(|\vec{a} \times \vec{b}|\), if \(\vec{a}\) = 2i? + j? + 5k? and \(\vec{b}\) = 3i? + 5j? – 2k?. (All India 2019)

Question 7.

If i? + j? + k?, 2i? + 5j?, 5i? + 2j? – 5k? and i? – 6j? – k? respectively, are the position vectors of points A, B, C and D, then find the angle between the straight lines AB and CD. Find whether \(\overrightarrow{A B}\) and \(\overrightarrow{C D}\) are collinear or not. (Delhi 2019)

Question 8.

The scalar product of the vector \(\vec{a}\) = i? + j? + k? with a unit vector along the sum of the vectors \(\vec{b}\) = 2i? + 4j? – 5k? and \(\vec{c}\) = ?i? + 2j? + 5k? is equal to 1. Find the value of ? and hence find the unit vector along \(\vec{b}\) + \(\vec{c}\). (All India 2019)

Question 9.

Find the magnitude of each of the two vectors \(\vec{a}\) and \(\vec{b}\), having the same magnitude such that the angle between them is 60° and their scalar product is \(\frac{9}{2}\). (CBSE 2018)

Question 10.

Find the value of [i?, k?, j?], (CBSE 2018C)

Question 1.

If ? is the angle between two vectors i? – 2 j? + 3k? and 3i? – 2 j? + k?, find sin ?. (CBSE 2018)

Question 2.

If \(\vec{a}+\vec{b}+\vec{c}\) = 0 and |\(\vec{a}\)| = 5, |\(\vec{b}\)| = 6 and |\(\vec{c}\)| = 9, then find the angle between \(\vec{a}\) and \(\vec{b}\). (CBSE 2018C)

Question 3.

Let \(\vec{a}\) = 4 i? + 5j? – k?, \(\vec{b}\) = i? – 4j? + 5k? and \(\vec{c}\) = 3i? + j? – k?. Find a vector which is perpendicular to both \(\vec{c}\) and \(\vec{b}\) and \(\vec{d} \cdot \vec{a}\) = 21. (CBSE 2018)

Question 4.

Find x such that the four points A(4, 4, 4), B(5, x, 8), C(5, 4, 1) and D (7, 7, 2) are coplanar. (CBSE 2018C)

Question 5.

Find the value of x such that the points A(3, 2, 1), B(4, x, 5), C(4, 2,- 2) and D (6, 5, -1) are coplanar. (All India 2017)

Question 6.

If \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are three mutually perpendicular vectors of the same magnitude, then prove that \(\vec{a}+\vec{b}+\vec{c}\) is equally inclined with the vectors \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\). (Delhi 2017, 2013C, 2011)

Question 7.

Using vectors, find the area of the ?ABC, whose vertices are A(1, 2, 5), 5(2, -1, 4) and C(4, 5, -1). (Delhi 2017; All India 2013)

Question 8.

Let \(\vec{a}\) = i? + j? + k?, \(\vec{b}\) = i? + 0 ? j? + 0 ? k? and \(\vec{c}\) = c₁i? + c₂j? + c₃k?, then
(a) Let c₁ = 1 and c₂ = 2, find c₃ which makes \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) coplanar.
(b) If c₂ = – 1 and c₃ = 1, show that no value of c₁ can make \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) coplanar. (Delhi 2017)

Question 9.

Show that the points A, B, C with position vectors 2i? – j? + k?, i? – 5j? – 5k? and 5i? – 4j? – 4k? respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle. (All India 2017)

Question 10.

Find the position vector of a point which divides the join of points with position vectors \(\vec{a}-2 \vec{b}\) and 2\(2 \vec{a}+\vec{b}\) externally in the ratio 2:1. (Delhi 2016)