91Ó°ÊÓ

(a) The equation \(x+y=1\) can be viewed as a linear system of one equation in two unknowns. Express a general solution of this equation as a particular solution plus a general solution of the associated homogeneous system. (b) Give a geometric interpretation of the result in part (a).

For what value(s) of \(t,\) if any, is the given vector parallel to \(\mathbf{u}=(4,-1) ?\) (a) \((8 t,-2)\) (b) \((8 t, 2 t)\) (c) \(\left(1, t^{2}\right)\)

Find the vector component of \(u\) along a and the vector component of \(u\) orthogonal to a. $$\mathbf{u}=(-1,-2), \mathbf{a}=(-2,3)$$

Compute the scalar triple product \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})\). $$\mathbf{u}=(-1,2,4), \quad \mathbf{v}=(3,4,-2), \quad \mathbf{w}=(-1,2,5)$$

Find the cosine of the angle \(\theta\) between \(\mathbf{u}\) and \(\mathbf{v}\). (a) \(\mathbf{u}=(2,3), \quad \mathbf{v}=(5,-7)\) (b) \(\mathbf{u}=(-6,-2), \quad \mathbf{v}=(4,0)\) (c) \(\mathbf{u}=(1,-5,4), \quad \mathbf{v}=(3,3,3)\) (d) \(\mathbf{u}=(-2,2,3), \quad \mathbf{v}=(1,7,-4)\)

Find the vector component of \(u\) along a and the vector component of \(u\) orthogonal to a. $$\mathbf{u}=(3,1,-7), \mathbf{a}=(1,0,5)$$

Which of the following vectors in \(R^{6}\) are parallel to \(\mathbf{u}=(-2,1,0,3,5,1) ?\) (a) (4,2,0,6,10,2) (b) (4,-2,0,-6,-10,-2) (c) (0,0,0,0,0,0)

Compute the scalar triple product \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})\). $$\mathbf{u}=(a, 0,0), \quad \mathbf{v}=(0, b, 0), \quad \mathbf{w}=(0,0, c)$$

(a) Find a homogeneous linear system of two equations in three unknowns whose solution space consists of those vectors in \(R^{3}\) that are orthogonal to \(a=(1,1,1)\) and \(b=(-2,3,0)\) (b) What kind of geometric object is the solution space? (c) Find a general solution of the system obtained in part (a), and confirm that Theorem 3.4 .3 holds.

Compute the scalar triple product \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})\). $$\mathbf{u}=(3,-1,6), \quad \mathbf{v}=(2,4,3), \quad \mathbf{w}=(5,-1,2)$$