List of top Mathematics Questions on Vector Algebra asked in UP Board XII

Three vectors \(\vec a, \vec b\) and \(\vec c\) satisfy the condition \(\vec a+\vec b+\vec c=0\). If \(|\vec a|=3, |\vec b|=4\) and \(|\vec c|=5\), then find the value of \(\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a\).

Prove that \((\vec a+\vec b)\cdot(\vec a+\vec b) = |\vec a|^2+|\vec b|^2\) if and only if \(\vec a\) and \(\vec b\) are perpendicular. It is given that \(\vec a\ne0,\ \vec b\ne0\).

Find the area of the parallelogram whose adjacent sides are given by \(\vec a = 3\hat i+\hat j+4\hat k\) and \(\vec b = \hat i-\hat j+\hat k\).

Let three vectors \(\vec a,\vec b\) and \(\vec c\) be such that \(|\vec a|=3,|\vec b|=4,|\vec c|=5\) and each of them is perpendicular to the sum of the other two vectors, then find the value of \(|\vec a+\vec b+\vec c|\).

If the position vectors of the vertices \(A,B,C\) of a \(\triangle ABC\) are respectively \(\vec a,\vec b\) and \(\vec c\), then prove that the area of \(\triangle ABC\) will be \(\dfrac12\left|\vec a\times \vec b+\vec b\times \vec c+\vec c\times \vec a\right|\).

Show that the points \(A,B,C\), whose position vectors are respectively \(\vec a=3\hat i-4\hat j-4\hat k\), \(\vec b=2\hat i-\hat j+\hat k\) and \(\vec c=\hat i-3\hat j-5\hat k\), form the vertices of a right-angled triangle.

For which value of \(\lambda\) is the vector \(2\hat{i}+\hat{j}-\lambda\hat{k}\) perpendicular to the vector \(\hat{i}+4\hat{j}+\hat{k}\)?

Prove that vectors \(2\hat i-\hat j+\hat k\), \(\hat i-3\hat j-5\hat k\) and \(3\hat i-4\hat j-4\hat k\) are vertices of a right-angled triangle.

Find the projection of vector \(\hat i+3\hat j+7\hat k\) on vector \(7\hat i-\hat j+8\hat k\).

Find the angle between vectors \(\hat i-2\hat j+3\hat k\) and \(3\hat i-2\hat j+\hat k\).

If vectors \(5\hat i-\lambda\hat j+2\hat k\) and \(2\hat i+3\hat j+4\hat k\) are perpendicular to each other, then the value of \(\lambda\) is:

If vectors \(\vec a,\vec b,\vec c\) are mutually perpendicular and having equal magnitude, then show that the vector \(\vec a+\vec b+\vec c\) is equally inclined to the vectors \(\vec a,\vec b,\vec c\).

Find the magnitude of the vectors \(\vec a\) and \(\vec b\), if having equal magnitudes and angle \(60^\circ\) between them and their scalar product is \(\dfrac12\).

Find the projection of vector \((\hat i-\hat j)\) on vector \((\hat i+\hat j)\).

Three vectors \(\vec a,\vec b,\vec c\) satisfy the condition \(\vec a+\vec b+\vec c=0\). If \(|\vec a|=3,|\vec b|=4\) and \(|\vec c|=2\), then find the value of \(\mu=\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a\).

If \((\vec a+\vec b)\cdot(\vec a-\vec b)=8\) and \(|\vec a|=8|\vec b|\), then find \(|\vec a|\) and \(|\vec b|\).

If two vectors \(\vec a=5\hat i-\hat j-3\hat k\) and \(\vec b=\hat i+3\hat j-5\hat k\), then show that \((\vec a+\vec b)\) and \((\vec a-\vec b)\) are perpendicular.

Find the angle between the vectors \(\vec a=\hat i+\hat j-\hat k\) and \(\vec b=\hat i-\hat j+\hat k\).

Suppose that \(\vec a\), \(\vec b\) and \(\vec c\) are three vectors such that \(|\vec a|=3\), \(|\vec b|=4\), \(|\vec c|=5\), and each of them is perpendicular to the sum of the other two vectors. Find \(|\vec a+\vec b+\vec c|\).

If the position vectors of points \(A\), \(B\), \(C\) and \(D\) are respectively \(\hat i+\hat j+\hat k\), \(2\hat i+5\hat j\), \(3\hat i+2\hat j-3\hat k\) and \(\hat i-6\hat j-\hat k\), then find the angle between the lines \(AB\) and \(CD\). Prove that \(AB\) and \(CD\) are collinear.

If for a unit vector \(\vec a\), \((\vec x-\vec a)\cdot(\vec x+\vec a)=12\), then find \(|\vec x|\).

If \(\vec a=2\hat i+\hat j+3\hat k\) and \(\vec b=3\hat i+5\hat j-2\hat k\), find \(|\vec a\times\vec b|\).

The value of expression \(\hat{i} \cdot \hat{i} - \hat{j} \cdot \hat{j} + \hat{k} \times \hat{k}\) is