Vector Algebra - Algebra - Mathematics Previous Year Questions

If $A, B, C$ and $D$ are $(3,7,4),(5,-2,-3),(-4,5,6)$ and $(1,2,3)$ respectively, then the volume of the parallelopiped with $A B, A C$ and $A D$ as the co-terminus edges, is .......... cubic units.

If the vectors $x \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+y \hat{\mathbf{j}}-z \hat{\mathbf{k}}$ are collinear then the value of $\frac{x y^2}{z}$ is equal

Which of the following is not equal to $\mathbf{w} \cdot(\mathbf{u} \times \mathbf{v})$ ?

If $\mathbf{a}+\mathbf{b}, \mathbf{b}+\mathbf{c}$ anc $\mathbf{c}+\mathbf{a}$ are coterminous edges of a parallel opiped then its volume is ..........

If $\mathbf{p}, \mathbf{q}$ and $\mathbf{r}$ are non-zero, non-coplanar vectors

For any non zero vector, a, b, c $\mathbf{a} \cdot[(\mathbf{b}+\mathbf{c}) \times(\mathbf{a}+\mathbf{b}+\mathbf{c})]=$ ..........

If the scalar triple product of the vectors $-3 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, 3 \hat{\mathbf{i}}-7 \hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}$ and $7 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ is 272 then $\lambda=\ldots \ldots$

$\mathbf{a}$ and $\mathbf{b}$ are non-collinear vectors. If $\mathbf{c}=(x-2) \mathbf{a}+\mathbf{b}$ and $\mathbf{d}=(2 x+1) \mathbf{a}-\mathbf{b}$ are collinear vectors, then the value of $x=\ldots \ldots$

Let \(G\) be the centroid of a \(\triangle A B C\) and \(\mathrm{O}_{b_\theta}\) other point in that plane, then \(\mathrm{OA}+\mathrm{OB}+\mathrm{OC}+\mathrm{CG}=\)

For any non-zero vectors \(\mathbf{a}\) and \(\mathbf{b}\),

If the volume of the parallelopiped whose conterminus edges are along the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) is 12, then the volume of the tetrahedron whose conterminus edges are \(\mathbf{a}+\mathbf{b}, \mathbf{b}+\mathbf{c}\) and \(c+a\) is

In a quadrilateral \(ABCD, M\) and \(N\) are the mid-points of the sides \(A B\) and \(C D\) respectively. If \(\mathbf{A D}+\mathbf{B C}=t \mathbf{M N}\), then \(t=\)

If the vectors \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+m \hat{\mathbf{k}}\) are coplanar, then \(m=\)

The angles between the lines \(\mathbf{r}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}) \text { and } \mathbf{r}=(3 \hat{\mathbf{i}}+\hat{\mathbf{k}})+\lambda^{\prime}(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}), \lambda, \lambda^{\prime} \in \mathbf{R}\) is

If \([\vec{a}\ \vec{b}\ \vec{c}\ ] \neq 0\), then \(\frac{[\vec{a}\ +\vec{b}\ \vec{b}\ +\vec{c}\ \vec{c}\ +\vec{a}\ ]}{[\vec{b}\ \vec{c}\ \vec{a}\ ]}=\)

$\mathbf{a}$ and $\mathbf{b}$ are non-collinear vectors. If $p=(2 x+1) a-b$ and $q=(x-2) a+b$ are collinear vectors, then $x=$

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are non-coplanar vectors and $p=\frac{\mathbf{b} \times \mathbf{c}}{[a b c]}, q=\frac{\mathbf{c} \times \mathbf{a}}{[a b c]}, r=\frac{\mathbf{a} \times \mathbf{b}}{[a b c]}$, then $\mathbf{a} \cdot \mathbf{p}+\mathbf{b} \cdot \mathbf{q}+\mathbf{c} \cdot \mathbf{r}=$

If $\mathbf{a}=3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}, \mathbf{b}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+7 \hat{\mathbf{k}}$ and $\mathbf{c}=7 \hat{\mathbf{i}}-\hat{\mathbf{j}}+23 \hat{\mathbf{k}}$ are three vectors, then which of the following statement is true.

\(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are vectors such that \(|\overline{\mathrm{a}}|=5,|\overline{\mathrm{b}}|=4,|\overline{\mathrm{c}}|=3\) and each is perpendicular to the sum of the other two, then \(|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|^2=\)

\(\overline{\mathrm{a}}, \overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\) are three vectors such that \(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}=\overline{0}\) and \(|\overline{\mathrm{a}}|=3,|\overline{\mathrm{b}}|=5,|\overline{\mathrm{c}}|=7\), then the angle between \(\overline{\mathrm{a}}\) and \(\bar{b}\) is